ECSS-E-ST-60-20C Rev. 1
15 November 2008
Space engineering
Stars sensors terminology and
performance specification
ECSS Secretariat
ESA-ESTEC
Requirements & Standards Division
Noordwijk, The Netherlands
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Foreword
This Standard is one of the series of ECSS Standards intended to be applied together for the
management, engineering and product assurance in space projects and applications. ECSS is a
cooperative effort of the European Space Agency, national space agencies and European industry
associationsforthepurposeofdevelopingand
maintainingcommonstandards.Requirementsinthis
Standardaredefinedintermsofwhatshallbeaccomplished,ratherthanintermsofhowtoorganize
and perform the necessary work. This allows existing organizational structures and methods to be
appliedwherethey are effective, and for thestructuresand methods to evolve
as necessarywithout
rewritingthestandards.
This Standard has been prepared by the ECSSEST6020 Working Group, reviewed by the ECSS
ExecutiveSecretariatandapprovedbytheECSSTechnicalAuthority.
Disclaimer
ECSSdoesnotprovideanywarrantywhatsoever,whetherexpressed,implied,orstatutory,including,
butnotlimitedto,
anywarrantyofmerchantabilityorfitnessforaparticularpurposeoranywarranty
that the contents of the item are errorfree. In no respect shall ECSS incur any liability for any
damages,including,butnotlimitedto, direct, indirect,special,orconsequentialdamagesarisingout
of, resulting from, or
in any way connected to the use of this Standard, whether or not based upon
warranty, business agreement,tort, or otherwise; whether ornot injury wassustained bypersons or
propertyorotherwise;andwhetherornotlosswassustainedfrom,oraroseoutof,theresultsof,the
item,or
anyservicesthatmaybeprovidedbyECSS.
Publishedby: ESARequirementsandStandardsDivision
ESTEC, P.O. Box 299,
2200 AG Noordwijk
The Netherlands
Copyright: 2008 © by the European Space Agency for the members of ECSS
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Change log
ECSSEST6020A Neverissued
ECSSEST6020B Neverissued
ECSSEST6020C
31July2008
Firstissue
ECSSEST6020CRev.1
15November2008
Firstissuerevision1.
ChangeswithrespecttoversionC(31July2008)areidentifiedwith
revisiontracking.
Mainchangesare:
Theterm“imaginaryensemble”hasbeenreplacedinthewholedocument
with“statisticalensemble”tobeconsistentwithECSSE
ST6010.
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Table of contents
1 Scope.......................................................................................................................9
2 Normative references...........................................................................................10
3 Terms, definitions and abbreviated terms..........................................................11
3.1 Terms from other standards .....................................................................................11
3.2 Terms specific to the present standard ....................................................................11
3.3 Abbreviated terms ....................................................................................................30
4 Functional requirements......................................................................................32
4.1 Star sensor capabilities ............................................................................................32
4.1.1 Overview.....................................................................................................32
4.1.2 Cartography................................................................................................33
4.1.3 Star tracking ...............................................................................................34
4.1.4 Autonomous star tracking...........................................................................34
4.1.5 Autonomous attitude determination............................................................35
4.1.6 Autonomous attitude tracking.....................................................................36
4.1.7 Angular rate measurement.........................................................................36
4.1.8 (Partial) image download............................................................................37
4.1.9 Sun survivability..........................................................................................37
4.2 Types of star sensors...............................................................................................38
4.2.1 Overview.....................................................................................................38
4.2.2 Star camera................................................................................................38
4.2.3 Star tracker.................................................................................................38
4.2.4 Autonomous star tracker ............................................................................38
4.3 Reference frames.....................................................................................................39
4.3.1 Overview.....................................................................................................39
4.3.2 Provisions...................................................................................................39
4.4 On-board star catalogue...........................................................................................39
5 Performance requirements..................................................................................41
5.1 Use of the statistical ensemble.................................................................................41
5.1.1 Overview.....................................................................................................41
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5.1.2 Provisions...................................................................................................41
5.2 Use of simulations in verification methods ...............................................................42
5.2.1 Overview.....................................................................................................42
5.2.2 Provisions for single star performances .....................................................42
5.2.3 Provisions for quaternion performances.....................................................42
5.3 Confidence level.......................................................................................................42
5.4 General performance conditions ..............................................................................43
5.5 General performance metrics...................................................................................44
5.5.1 Overview.....................................................................................................44
5.5.2 Bias.............................................................................................................44
5.5.3 Thermo elastic error ...................................................................................45
5.5.4 FOV spatial error ........................................................................................46
5.5.5 Pixel spatial error........................................................................................47
5.5.6 Temporal noise...........................................................................................47
5.5.7 Aberration of light .......................................................................................48
5.5.8 Measurement date error.............................................................................49
5.5.9 Measured output bandwidth .......................................................................49
5.6 Cartography..............................................................................................................49
5.7 Star tracking .............................................................................................................49
5.7.1 Additional performance conditions .............................................................49
5.7.2 Single star tracking maintenance probability..............................................50
5.8 Autonomous star tracking.........................................................................................50
5.8.1 Additional performance conditions .............................................................50
5.8.2 Multiple star tracking maintenance level.....................................................50
5.9 Autonomous attitude determination..........................................................................51
5.9.1 General.......................................................................................................51
5.9.2 Additional performance conditions .............................................................51
5.9.3 Verification methods...................................................................................52
5.9.4 Attitude determination probability ...............................................................52
5.10 Autonomous attitude tracking...................................................................................53
5.10.1 Additional performance conditions .............................................................53
5.10.2 Maintenance level of attitude tracking ........................................................54
5.10.3 Sensor settling time....................................................................................55
5.11 Angular rate measurement.......................................................................................55
5.11.1 Additional performance conditions .............................................................55
5.11.2 Verification methods...................................................................................55
5.12 Mathematical model .................................................................................................56
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Annex A (normative) Functional mathematical model (FMM) description -
DRD ......................................................................................................................57
Annex B (informative) Ancillary terms in Star Sensors .......................................59
Annex C (informative) Optional features of star sensors ....................................68
Annex D (informative) Performance metrics applied to star sensors.................72
Annex E (informative) Statistics.............................................................................74
Annex F (informative) Transformations between coordinate frames .................80
Annex G (informative) Contributing Error Sources..............................................82
Annex H (informative) Example of data sheet.......................................................84
Figures
Figure 3-1: Star sensor elements – schematic.......................................................................14
Figure 3-2: Example alignment reference frame....................................................................16
Figure 3-3: Boresight reference frame ...................................................................................17
Figure 3-4: Example of Inertial reference frame.....................................................................17
Figure 3-5: Mechanical reference frame ................................................................................18
Figure 3-6: Schematic illustration of reference frames...........................................................19
Figure 3-7: Stellar reference frame ........................................................................................19
Figure 3-8: Schematic timing diagram....................................................................................21
Figure 3-9: Field of View ........................................................................................................23
Figure 3-10: Aspect angle to planetary body or sun...............................................................24
Figure 4-1: Schematic generalized Star Sensor model..........................................................33
Figure B-1 : AME, MME schematic definition.........................................................................63
Figure B-2 : RME Schematic Definition..................................................................................64
Figure B-3 : MDE Schematic Definition..................................................................................65
Figure B-4 : Rotational and directional Error Geometry .........................................................66
Figure F-1 : Angle rotation sequence.....................................................................................81
Figure H-1 : Example of detailed data sheet..........................................................................85
Tables
Table C-1 : Minimum and optional capabilities for star sensors.............................................71
Table D-1 : Measurement error metrics .................................................................................73
Table D-2 : Star Position measurement error metrics............................................................73
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Table E-1 : Minimum number of simulations to verify a performance at performance
confidence level P
C
to an estimation confidence level of 95 %............................78
Table G-1 : Contributing error sources...................................................................................82
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Introduction
Inrecentyearstherehave been rapiddevelopmentsinstartrackertechnology,
in particular with a great increase in sensor autonomy and capabilities. This
Standard is intended to support the variety of star sensors either available or
underdevelopment.
This Standard defines the terminology and specification definitions for the
performanceofstartrackers(inparticular,autonomousstartrackers).Itfocuses
on the specific issues involved in the specification of performances of star
trackersandisintendedtobeusedasastructuredsetofsystematicprovisions.
This Standard is not intended to replace textbook material on star tracker
technology,andsuchmaterialisintentionallyavoided.Thereadersandusersof
this Standard are assumed to possess general knowledge of star tracker
technologyanditsapplicationtospacemissions.
This document defines and normalizes terms used in star sensor performance
specifications,aswellassomeperformanceassessmentconditions:
sensorcomponents
sensorcapabilities
sensortypes
sensorreferenceframes
sensormetrics
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1
Scope
ThisStandardspecifiesstartrackerperformancesaspartofaspaceproject.The
Standard covers all aspects of performances, including nomenclature,
definitions, and performance metrics for the performance specification of star
sensors.
The Standard focuses on performancespecifications.Other specificationtypes,
for example mass and power, housekeeping data, TM/TC interface and
data
structures,areoutsidethescopeofthisStandard.
When viewed from the perspective of a specific project context, the
requirementsdefinedinthisStandardshouldbetailoredto match thegenuine
requirementsofaparticularprofileandcircumstancesofaproject.
Thisstandardmaybetailoredforthespecific
characteristicsandconstraintsofa
spaceprojectinconformancewithECSSSST00.
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2
Normative references
The following normative documents contain provisions which, through
reference in this text, constitute provisions of this ECSS Standard. For dated
references,subsequentamendmentsto,orrevisionofanyofthesepublications,
donotapply.However,partiestoagreementsbasedonthisECSSStandardare
encouragedtoinvestigatethepossibilityofapplying
themorerecenteditionsof
the normative documents indicated below. For undated references, the latest
editionofthepublicationreferredtoapplies.
ECSSSST0001 ECSSsystemGlossaryofterms
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3
Terms, definitions and abbreviated terms
3.1 Terms from other standards
ForthepurposeofthisStandard,thetermsanddefinitionsfromECSSSST0001
apply.Additionaldefinitionsareincludedin
AnnexB.
3.2 Terms specific to the present standard
3.2.1 Capabilities
3.2.1.1 aided tracking
capability to input information to the star sensor internal processing from an
externalsource
NOTE1 This capability applies to star tracking,
autonomous star tracking and autonomous
attitudetracking.
NOTE2 E.g.AOCS.
3.2.1.2 angular rate measurement
capability to determine, the instantaneous sensor reference frame inertial
angularrotationalrates
NOTE Angularratecanbecomputedfromsuccessivestar
positions obtained from the detector or successive
absoluteattitude (derivationof successive
attitude).
3.2.1.3 autonomous attitude determination
capability to determine the absolute orientation of a defined sensor reference
frame with respect to a defined inertial reference frame and to do so without
the use of any a priori or externally supplied attitude, angularrate or angular
accelerationinformation
3.2.1.4 autonomous attitude tracking
capabilitytorepeatedlyreassessandupdatetheorientationofasensordefined
reference frame with respect to an inertially defined reference frame for an
extended period of time, using autonomously selected star images in the field
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of view, following the changing orientation of the sensor reference frame as it
movesinspace
NOTE1 TheAutonomousAttitudeTrackingmakesuseofa
supplied a priori Attitude Quaternion, either
provided by an external source (e.g. AOCS) or as
the output of an Autonomous Attitude
Determination(‘LostinSpace’
solution).
NOTE2 The autonomous attitude tracking functionality
can also be achieved by the repeated use of the
AutonomousAttitudeDeterminationcapability.
NOTE3 The Autonomous Attitude Tracking capability
does not imply the solution of the ‘lost in space’
problem.
3.2.1.5 autonomous star tracking
capabilitytodetect,locate,selectandsubsequentlytrackstarimageswithinthe
sensorfieldofview foranextended period oftimewith no assistanceexternal
tothesensor
NOTE1 Furthermore, the autonomous star tracking
capability is taken to include the ability to
determinewhenatrackedimageleavesthe
sensor
fieldofviewand select a replacementimage to be
trackedwithoutanyuserintervention.
NOTE2 Seealso
3.2.1.9(startracking).
3.2.1.6 cartography
capability to scan the entire sensor field of view and to locate and output the
positionofeachstarimagewithinthatfieldofview
3.2.1.7 image download
capabilitytocapturethesignalsfromthedetectorovertheentiredetectorField
of view, at one instant (i.e. within a single integration), and output all of that
informationtotheuser
NOTE Seealso
3.2.1.8(partialimagedownload).
3.2.1.8 partial image download
capabilitytocapturethesignalsfromthedetectorovertheentiredetectorField
ofview,atone instant(i.e.withinasingleintegration),andoutput part ofthat
informationtotheuser
NOTE1 Partial image download is an image downloads
(see
3.2.1.7)whereonlyapartofthedetectorfield
of view can be output for any given specific
‘instant’.
NOTE2 Partial readout of the detector array (windowing)
andoutput of thecorrespondingpixelsignalsalso
fulfilthefunctionality.
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3.2.1.9 star tracking
capability to measure the location of selected star images on a detector, to
output the coordinates of those star images with respect to a sensor defined
reference frame and to repeatedly reassess and update those coordinates for
an extended period of time, following the motion of each image
across the
detector
3.2.1.10 sun survivability
capability to withstand direct sun illumination along the boresight axis for a
certain period of time without permanent damage or subsequentperformance
degradation
NOTE This capability could be extended to flare
capability considering the potential effect of the
earthorthemoonintheFOV.
3.2.2 Star sensor components
3.2.2.1 Overview
Figure31showsaschemeoftheinterfaceamongthegeneralized components
specifiedinthisStandard.
NOTE Usedasacamerathesensoroutputcanbelocated
directlyafterthepreprocessingblock.
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BAFFLE
OPTICAL
HEAD
OPTICAL SYSTEM
PROCESSOR
PROCESS OUPUT
DETECTOR
MEMORY
CAMERA
OUTPUT
PRE-PROCESSING
Figure31:Starsensorelementsschematic
3.2.2.2 baffle
passive structure used to prevent or reduce the entry into the sensor lens or
aperture of any signals originating from outside of the
field of view of the
sensor
NOTE Baffle design is usually mission specific and
usually determines the effective exclusion angles
forthelimboftheEarth,MoonandSun.TheBaffle
canbemounted directly onthesensoror can be a
totally separate element. In the latter case, a
positioningspecificationwithrespecttothesensor
isused.
3.2.2.3 detector
element of the star sensor that converts the incoming signal (photons) into an
electricalsignal
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NOTE Usual technologies in use are CCD (charge
coupled device) and APS (active pixel sensor)
arrays though photomultipliers and various other
technologiescanalsobeused.
3.2.2.4 electronic processing unit
setoffunctionsofthesensornotcontainedwithintheopticalhead
NOTE Specifically,thesensorelectronicscontains:
sensorprocessor;
powerconditioning;
softwarealgorithms;
onboardstarcatalogue(ifpresent).
3.2.2.5 optical head
partofthesensorresponsibleforthecaptureandmeasurementoftheincoming
signal
NOTE Assuchitconsistsof
theopticalsystem;
thedetector(includinganycoolingequipment);
the proximity electronics (usually detector
control, readout and interface, and optionally
pixelpreprocessing);
themechanicalstructure
tosupporttheabove.
3.2.2.6 optical system
systemthat comprises the componentparts tocapture andfocusthe incoming
photons
NOTE Usually this consists of a number of lenses, or
mirrorsandfilters,andthesupportingmechanical
structure,stops,pinholesandslitsifused.
3.2.3 Reference frames
3.2.3.1 alignment reference frame (ARF)
referenceframefixedwithrespecttothesensorexternalopticalcubewherethe
origin of the ARF is defined unambiguously with reference to the sensor
externalopticalcube
NOTE1 The X, Y‐ and Zaxes of the ARF are a right
handed orthogonal set of axes which are defined
unambiguously with respect to the normal of the
faces of the external optical cube.
Figure 32
schematicallyillustratesthedefinitionoftheARF.
NOTE2 The ARF is the frame used to align the sensor
duringintegration.
NOTE3 This definition does not attempt to prescribe a
definitionoftheARF,otherthanitisaframefixed
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relative to the physical geometry of the sensor
opticalcube.
NOTE4 If the optical cube’s faces are not perfectly
orthogonal, the Xaxis can be defined as the
projection of the normal of the Xfacein the plane
orthogonaltotheZaxis,and the Yaxis completes
the
RHS.
Optical
Cube
X
ARF
Y
ARF
Z
ARF
Sensor
Figure32:Examplealignmentreferenceframe
3.2.3.2 boresight reference frame (BRF)
referenceframewhere:
the origin of the Boresight Reference Frame (BRF)is defined
unambiguously with reference to the mounting interface plane of the
sensorOpticalHead;
NOTE In an ideally aligned optoelectrical system this
results in a measured position at the centre of the
detector.
the Z
axis of the BRF is defined to beantiparallel to the direction of an
incomingcollimatedlightraywhichisparalleltotheopticalaxis;
XBRFaxis isin the plane spanned by ZBRFaxis and the vector from
the detector centre pointing along the positively counted
detector rows,
as the axis perpendicular to ZBRFaxis. The YBRFaxis completes the
righthandedorthogonalsystem.
NOTE1 TheXaxesandYaxesoftheBRFaredefinedtolie
(nominally) in the plane of the detector
perpendicular to the Zaxis, so as to
form a right
handed set with one axis nominally along the
detector array row and the other nominally along
thedetectorarraycolumn.
Figure33schematically
illustratesthedefinitionoftheBRF.
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NOTE2 The definition of the Boresight Reference Frame
does not imply that it is fixed with respect to the
Detector, but that it is fixed with respect to the
combineddetectorandopticalsystem.
Optics
Detector
Z
BRF
Y
BRF
X
BRF
Incoming light ray that
will give a measured
position at the centre of
the Detector.
Figure33:Boresightreferenceframe
3.2.3.3 inertial reference frame (IRF)
referenceframedeterminedtoprovideaninertialreference
NOTE1 E.g.usetheJ2000referenceframeasIRFasshown
in
Figure34.
NOTE2 TheJ2000referenceframe
(inshortforICRFInertial
Celestial Reference Frame at J2000 Julian date)
is
usually defined as Z IRF = earth axis of rotation
(direction of north) at J2000 (01/01/2000 at noon
GMT),XIRF=directionofvernalequinoxatJ2000,
Y IRF completes the righthanded orthonormal
referenceframe.
Ecliptic Plane
Equatorial Plane
X
IRF
Y
IRF
Z
IRF
at J2000 Julian date
X-axis in direction of
vernal equinox
ϒ
Earth
Figure34:ExampleofInertialreferenceframe
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3.2.3.4 mechanical reference frame (MRF)
reference frame where the origin of the MRF is defined unambiguously with
referencetothemountinginterfaceplaneofthesensorOpticalHead
NOTE1 For Fused Multiple Optical Head configurations,
the interface plane of one of the Optical Heads
may be nominated to define the MRF. The
orientationisto
bedefined.
NOTE2 E.g. the Zaxis of the MRF is defined to be
perpendiculartothemountinginterfaceplane.The
X‐andYaxesof the MRF are defined tolieinthe
mounting plane such as to form an orthogonal
RHSwiththeMRFZaxis.
NOTE
3 Figure 35 schematically illustrates the definition
oftheMRF.
Y
MRF
X
MRF
Spacecraft Body
Mounting Interface
Z
MRF
Figure35:Mechanicalreferenceframe
3.2.3.5 stellar reference frame (SRF)
reference frame for each star where the origin of any SRF is defined to be
coincidentwiththeBoresightReferenceFrame(BRF)origin
NOTE1 TheZaxisofanySRFisdefinedtobethedirection
from the SRF origin to the true position of the
selected star
Figure 36 gives a schematic
representation of the reference frames.
Figure 37
schematicallyillustratesthedefinitionoftheSRF.
NOTE2 The X‐ and Y‐ axes ofthe SRFare obtained under
the assumption that the BRF can be brought into
coincidencewiththeSRFbytworotations,thefirst
aroundthe BRF Xaxisandthesecond around the
newBRFYaxis (which iscoincidentwith the SRF
Yaxis).
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Z
MRF
Optical
Cube
Spacecraft Body
Z
BRF
Sensor
Y
BRF
X
BRF
Z
ARF
Z
SRF
Mounting Plate
IRF Axes
Figure36:Schematicillustrationofreferenceframes
Y
SRF
X
SRF
X
BRF
Detector
Selected star
Z
SRF
Z
BRF
Y
BRF
1
s
t
rotation
2
n
d
rotation
Figure37:Stellarreferenceframe
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3.2.4 Definitions related to time and frequency
3.2.4.1 integration time
exposuretimeoverwhichphotonswerecollectedinthedetectorarraypriorto
readoutandprocessingtogeneratetheoutput(starpositionsorattitude)
NOTE1 Integrationtime can befixed, manually adjustable
orautonomouslyset.
NOTE2
Figure 38 illustrates schematically the various
times defined together with their inter
relationship.Thefigureincludesdatabeingoutput
from two Optical Heads, each of which is
separately processed prior to generation of the
sensor output. Note that for a Fused Multiple
Optical Head sensor; conceptually it is assumed
that
the filtered output is achieved via sequential
processingof datafrom a single head at a time as
the data is received.Hence, with this
understanding, the figure and the associated time
definitionsalsoapplytothissensorconfiguration.
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Integration
Processing
Output
Integration time
Optical
Head 1
Sample Time
Latency
Time data is
first available
PROCESSING
Optical Head
2
PROCESSING
OUT
Data is
accessed
OUT
Data
Flow
Time
Figure38:Schematictimingdiagram
3.2.4.2 measurement date
dateoftheprovidedmeasurement
NOTE1 Incaseofonboardfilteringthemeasurementdate
candeviatefromindividualmeasurementdates.
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NOTE2 Usually the midpoint of the integration time is
considered as measurement date for CCD
technology.
3.2.4.3 output bandwidth
maximumfrequencycontainedwithinthesensoroutputs
NOTE1 The bandwidth of the sensor is limited in general
byseveralfactors,including:
integrationtime;
samplingfrequency;
attitudeprocessingrate;
onboard filtering of data (in particular for
multipleheadunits).
NOTE2 The output bandwidth corresponds to the
bandwidthofthesensorseenasalowpassfilter.
3.2.5 Field of view
3.2.5.1 half-rectangular field of view
angular region around the Boresight Reference Frame (BRF) frame Zaxis,
specifiedbytheangularexcursionsaroundtheBRFX‐andYaxesbetweenthe
BRFZaxisandtheappropriaterectangleedge,withinwhichastarproducesan
imageontheDetectorarraythatisthenusedbythestar
sensor
NOTE1 ThisFieldofViewisdeterminedbytheopticsand
Detectordesign.Thisisschematicallyillustratedin
Figure39.
NOTE2 In the corners, the extent of the FOV for this
definition exceeds the quoted value (see
Figure
39).
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Detector
BRF Z axis
Light cone for Full
Cone Field of View
Light cone for
Half-Rectangular
Field of View
Full Cone Field
of View
Half Rectangular
Field of View
Figure39:FieldofView
3.2.5.2 full cone field of view
angular region around the Boresight Reference Frame (BRF) frame Zaxis,
specifiedasafullconeangle,withinwhichastarwillproduceanimageonthe
Detectorarraythatisthenusedbythestarsensor
NOTE ThisFieldofViewisdeterminedbytheopticsand
Detectordesign.
Thisisschematicallyillustratedin
Figure39.
3.2.5.3 pixel field of view
anglesubtendedbyasingleDetectorelement
NOTE PixelFieldofViewreplaces(andisidenticalto)the
commonlyusedtermInstantaneousFieldofView.
3.2.6 Angles of celestial bodies
3.2.6.1 aspect angle
halfcone angle between the Boresight Reference Frame (BRF) Zaxis and the
nearestlimbofacelestialbody
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Solar System
Body
Detector
Z
BRF
A
SPECT ANGL
E
(in plane of diagram)
MEA
SEA
EEA
Figure310:Aspectangletoplanetarybodyorsun
3.2.6.2 exclusion angle (EA)
lowestaspectangleofabodyatwhichquotedfullperformanceisachieved
NOTE1 The following particular exclusion angles can be
considered:
TheEarthexclusionangle(EEA),definedasthe
lowest aspect angle of fully illuminated Earth
(including the Earth atmosphere) at which
quoted full performance is achieved, as
shown
schematicallyin
Figure310.
The SunExclusionAngle (SEA),definedas the
lowest Aspect Angle of the Sun at which
quoted full performance is achieved, as shown
schematicallyin
Figure310.
TheMoonExclusionAngle(MEA)isdefinedas
the lowest Aspect Angle of the Full Moon at
which quoted full performance is achieved, as
shownschematicallyin
Figure310.
NOTE2 The value of any EA depends on the distance to
the object. In general, the bandwidth is the lowest
of the cutoff frequencies implied by the above
factors.
3.2.7 Most common terms
3.2.7.1 correct attitude
attitude for which the quaternion absolute measurementerror (AMEq defined
in
D.2.2)islowerthanagiventhreshold
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3.2.7.2 correct attitude threshold
maximum quaternion absolute measurement error (AMEq) for which an
attitudeisacorrectattitude
3.2.7.3 false attitude
attitudewhichisanoncorrectattitude
3.2.7.4 false star
signal on the detector not arising from a stellar source but otherwise
indistinguishablefromastarimage
NOTE This definition explicitly excludes effects from the
Moon, low incidence angle proton effects etc.,
whichcangenerallybedistinguishedasnonstellar
inoriginbygeometry.
3.2.7.5 image output time
timerequiredtooutputthedetectorimage
3.2.7.6 statistical ensemble
setofsensors(notallactuallybuilt)onwhichtheperformancesareassessedby
useofstatisticaltoolsonasetofobservationsandobservationconditions
NOTE1 The statistical
ensemble is defined on a caseby
case basis, depending on the performances to be
assessed.
NOTE2 See
5.1andAnnexEforfurtherdetails.
3.2.7.7 maintenance level of attitude tracking
total time within a longerdefinedintervalthat attitude tracking is maintained
(i.e. without any attitude acquisition being performed) with a probability of
100%foranyinitialpointingwithinthecelestialsphere
NOTE ThisparametercanalsobespecifiedasMeanTime
between loss of tracking or probability to
loose
trackingpertimeunit.
3.2.7.8 multiple star tracking maintenance level
total time within a longer defined interval that at least ‘n’ star tracks are
maintainedwithaprobabilityof100%
NOTE ThiscoversthecasewherethestarsintheFOVare
changing, such that the star tracks maintained
evolvewithtime.
3.2.7.9 night sky test
testperformedduringnighttimeusingtheskyasphysicalstimulusforthestar
sensor. The effect of atmospheric extinction should be taken into account and
reducedbyappropriatechoiceofthelocationfortest
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3.2.7.10 probability of correct attitude determination
probability that a correct attitude solution is obtained and is flagged as valid,
within a defined time from the start of attitude determination with the sensor
switchedonandattheoperatingtemperature
NOTE1 Time periods for other conditions, like recovery
aftertheSunenteringthe FOV or acold
start, can
be defined as the time needed to reach the start
time of the attitude determination. The total time
neededwouldthenbethesumofthetimeneeded
toreachthestarttimeoftheattitudedetermination
andthetimeperiodrelatedtothismetric.
NOTE2 Attitude
solution flagged as valid means that the
obtained attitude is considered by star sensor
suitable for use by the AOCS. The validity is
independentofaccuracy.
NOTE3 Correct attitude solution means that stars used to
derive the quaternion have been correctly
identified, i.e. error on delivered measurement is
belowa
definedthreshold.
3.2.7.11 probability of false attitude determination
probability that not correct attitude solution is obtained, which is flagged as
valid, within a defined time from the start of attitude determination with the
sensorswitchedonandattheoperatingtemperature
3.2.7.12 probability of invalid attitude solution
probabilitythatanattitudesolution(correctornotcorrect)isobtainedanditis
flagged as not valid, within a defined time from the start of attitude
determinationwiththesensorswitchedonandattheoperatingtemperature
NOTE1 The value of the Probability of Invalid Attitude
Solution is 1
(Probability of Correct Attitude
Determination + Probability of False Attitude
Determination).
NOTE2 Invalid attitude solutions include cases of silence
(i.e.noattitudeisavailablefromstarsensor).
3.2.7.13 sensor settling time
time period from the first quaternion output to the first quaternion at full
attitude accuracy, for random initial pointing within a defined region of the
celestialsphere
NOTE The time period is specified with a probability of
n%‐ifnotquoted,avalueof99%isassumed.
3.2.7.14 single star tracking maintenance probability
probabilitytobemaintainedbyanexistingstartrackoveradefinedtimeperiod
whilethetrackedstarisintheFOV
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3.2.7.15 star image
patternoflightfallingonthedetectorfromastellarsource
3.2.7.16 star magnitude
magnitudeofthestellarimageasseenbythesensor
NOTE Star magnitude takes into account spectral
considerations. This is also referred to as
instrumentalmagnitude.
3.2.7.17 validity
characteristics of an output of the star sensor being accurate enough for the
purposeitisintendedfor
NOTE E.g.usebytheAOCS.
3.2.8 Errors
3.2.8.1 aberration of light
Erroronthepositionofameasuredstarduetothetimeofpropagationoflight,
andthelinearmotionoftheSTRinaninertialcoordinatesystem
NOTE1 The Newtonian first order expression of the
rotationerrorforonestardirectionis:
()
usin
c
V
r
r
θ=ε
where:
V isthemagnitudeoftheabsolutelinearvelocity
V
r
ofthespacecraftw.r.t.toaninertialframe
c isthelightvelocity(299792458m/s)
 istheanglebetweenthe V
r
vectorandthestar
direction
n
r
nV
nV
u
r
r
r
r
r
=
NOTE2 For a satellite on an orbit around the Earth, the
absolute velocity is the vector sum of the relative
velocityofthespacecraftw.r.ttheEarthandofthe
velocityoftheEarthw.r.ttheSun.
NOTE3 For an Earth orbit, the magnitude of this effect is
around 25 arcsec (max). For an interplanetary
spacecraft the absolute velocity is simply the
absolutevelocityw.r.t.thesun.
NOTE4 The associated metrics is the MDE (see Annex
B.5.11 for the mathematical definition). The
detailed contributors to the relativistic error are
givenin
AnnexG.
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3.2.8.2 bias
errorontheknowledgeoftheorientationoftheBRFincluding:
the initial alignment measurement error between the Alignment
ReferenceFrame(ARF)andthe sensorBoresightReferenceFrame(BRF)
(ongroundcalibration)
theAlignmentStability Error(CalibrationtoFlight)witchisthechange
in the transformation between the sensor
Mechanical Reference Frame
(MRF)andthesensorBoresightReferenceFrame(BRF)betweenthetime
ofcalibrationandthestartoftheinflightmission
NOTE1 The bias can be for the BRF Zaxis directional or
therotationalerrorsaroundtheBRFX,Y‐axes.
NOTE2 For definition
of directional and rotational errors
see
B.5.14andB.5.17.
NOTE3 Duetoitsnature,thebiasmetricvalueisthesame
whatevertheobservationareais.
NOTE4 The associated metrics is the MME (see Annex
B.5.7forthemathematicaldefinition).Thedetailed
contributorstothebiasaregivenin
AnnexG.
3.2.8.3 FOV spatial error
error on the measured attitude quaternion due to theindividual spatial errors
onthestars
NOTE1 This error has a spatial periodicity, whose
amplitude is defined by the supplier. It ranges
fromafewpixelsuptothefullcameraFOV.
NOTE2 FOV spatial errors are mainly due to
optical
distortion. These errors can be converted to time
domain using sensor angular rate. Then, from
temporalfrequencypointofview,theyrangefrom
bias to high frequency errors depending on the
motion of stars on the detector. They lead to bias
error in the case of inertial pointing, while
they
contribute to random noise for high angular rate
missions.
NOTE3 The associated metrics is the MDE (see Annex
B.5.11 for the mathematical definition). The
detailed contributors to the FOV spatial error are
givenin
AnnexG.
3.2.8.4 pixel spatial error
Measurement errors of star positions due to detector spatial non uniformities
(including PRNU, DSNU, dark current spikes, FPN) and star centroid
computation(alsocalledinterpolationerror)
NOTE1 Becauseoftheir‘spatialnaturetheseerrorsvary
withthepositionofstarsonthedetectortheyare
well captured by
metrics working in the angular
domain. The pixel spatial errors are then well
defined as the errors on the measured attitude
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(respectively the measured star positions) due to
star measurement errors with spatial period of
TBD angular value. Several classes of spatial
periodscanbeconsidered.
NOTE2 These errors can be converted to time domain
using sensor angular rate. Then, from temporal
frequency point of view, they range from bias to
highfrequencyerrors dependingonthemotionof
starsonthedetector.Theyleadtobiaserrorinthe
case of inertial pointing, while they contribute to
randomnoiseforhighangularratemissions.
NOTE3 The associated metrics is the MDE (see Annex
B.5.11 for the mathematical definition). The
detailed contributors to the pixel spatial error are
givenin
AnnexG.
3.2.8.5 temporal noise
Temporal fluctuation on the measured quaternion (star positions) due to time
variationerrorsources
NOTE1 Temporalnoiseisawhitenoise.
NOTE2 TheassociatedmetricsistheRME(seeAnnex
B.5.8
for the mathematical definition). The detailed
contributors to the temporal noise error are given
in
AnnexG.
3.2.8.6 thermo elastic error
deviation of BRFversus MRF for a given temperature variation ofthe
mechanical interface of the optical head of the sensor and thermal power
exchangewithspace
NOTE1 The detailed contributors to the thermo elastic
erroraregivenin
AnnexG.
NOTE2 The associated metrics is the MDE (see Annex
B.5.11 for the mathematical definition). FOV
spatialerror.
3.2.9 Star sensor configurations
3.2.9.1 fused multiple optical head configuration
more than one Optical Head, each with a Baffle, and a single Electronic
ProcessingUnitproducingasinglesetofoutputsthatusesdatafromallOptical
Heads
3.2.9.2 independent multiple optical head configuration
more than one optical head, each with a baffle, and a single electronic
processingunitproducingindependentoutputsforeachopticalhead
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3.2.9.3 integrated single optical head configuration
singleopticalheadplusbaffleandasingleelectronicprocessingunitcontained
withinthesamemechanicalstructure
3.2.9.4 separated single optical head configuration
singleopticalheadplusbaffleandasingleelectronicprocessingunitwhichare
notcollocatedwithinthesamemechanicalstructure
3.3 Abbreviated terms
For the purpose of this Standard, theabbreviated terms from ECSSSST0001
andthefollowingapply:
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Abbreviation Meaning
AME
absolutemeasurementerror
APS
activepixelsensor
ARF
alignmentreferenceframe
ARME
absoluteratemeasurementerror
AST
autonomousstartracker
BRF
boresightreferenceframe
BOL
beginningoflife
CCD
chargecoupleddevice
CTE
chargetransferefficiency
DSNU
darksignalnonuniformity
EEA
Earthexclusionangle
EOL
endoflife
FMM
functionalmathematicalmodel
FOV
fieldofview
FPN
fixpatternnoise
GRME
generalizedrelativemeasurement
error
IRF
inertialreferenceframe
LOS
lineofsight
MDE
measurementdrifterror
MEA
Moonexclusionangle
MME
meanmeasurementerror
MRE
measurementreproducibilityerror
MRF
mechanicalreferenceframe
PRNU
photoresponsenonuniformity
RME
relativemeasurementerror
RHS
righthandedsystem
SEA
Sunexclusionangle
SEU
singeeventupset
SET
singleeventtransient
SRF
stellarreferenceframe
STC
starcamera
STM
starmapper
STR
startracker
STS
starscanner
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4
Functional requirements
4.1 Star sensor capabilities
4.1.1 Overview
This subclause describes the different main capabilities of star sensors. These
capabilities are defined with respect to a generalized description of the
reference frames (either sensorreferenced or inertially referenced in clause
3).
This set of capabilities is then later used to describe the specific types of star
sensorandtheirperformances.
In order to describe the star sensor capabilities, the following generalized
sensormodelisused:
A star sensor comprises an imaging function, a detecting function and a data
processingfunction. The
imaging function collects photons from objects in the
field of view of the sensor and focuses them on a detecting element. This
element converts the photons into an electrical signal that is then subject to
someprocessingtoproducethesensoroutput.
Aschematicofthissensormodelispresented
inFigure41.
For each capability the nominal outputs and additional outputs are defined.
Thesefunctionaldatashouldbeidentifiedinthetelemetrylistcomingfromthe
starsensor.
Theoutputsasdefinedinthisdocumentarepurelyrelatedtotheperformance
of the sensor, and represent the minimum information to
be provided by the
sensor to possess the capability. Other aspects, such as sensor housekeeping
data, data structures and the TM/TC interface, are outside the scope of this
Standard.
NOTE1 The same capabilities can be defined for Star
Sensors employed on spinning spacecraft (Star
Scanner)wherestarimagesareacquiredatangular
rateuptotensofdeg/sdrivingthedetectorwitha
dedicated technique. For Star Sensor based on
CCD detector, an example of this technique could
be the Time Delay Integration (TDI). It is outside
the scope of this specification to give detailed
capabilitydefinitionsforthiskindofsensor.
NOTE2 OptionalfeaturesareincludedinAnnex
B.6.
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Star
Light from star
Optics
Detector
Processor
Processed
output
Figure41:SchematicgeneralizedStarSensormodel
4.1.2 Cartography
4.1.2.1 Inputs
a. Theacquisitioncommandshallbesuppliedasminimumsetofinputs.
4.1.2.2 Outputs
a. Asensorwithcartographycapabilityshallhavethefollowingminimum
outputs:
1. starposition,
2. measurementdate.
b. WhentheStarImageismeasuredinaDetectorfixedframewhichisnot
the same as the Boresight Reference Frame (BRF), the output shall be
convertedintotheBoresightReferenceFrame
(BRF).
NOTE The output parameterization is the Star Image
position in the Boresight Reference Frame (BRF),
givenbythetwomeasuresoftheangularrotations
which define the transformation from the BRF to
thestarStellarReferenceFrame(SRF).
c. The date of measurement shall be expressed as a (scalar)
number
indicating the delay relative to a known external time reference agreed
withthecustomer.
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4.1.3 Star tracking
4.1.3.1 Inputs
a. The minimum set of inputs to be supplied in order to initializethe Star
Trackingshallbe:
1. theinitialstarposition;
2. theangularrate;
3. validitydate.
b. For aided tracking, data specified in
4.1.3.1ashall be supplied regularly
bythespacecraft,atanupdaterateandaccuracyagreedbythecustomer.
c. Theunitofallinputsshallbeindicated.
4.1.3.2 Outputs
a. A sensor with the star tracking capability shall have the following
minimumoutputs:
1. the position of each Star Image with respect to a sensordefined
referenceframe;
2. focallengthifstarpositiononthedetectorchipisoutputinunits
oflength;
3. themeasurementdate.
NOTE1
Theinitialselectionofthestarimagestobetracked
bythesensorisnotincludedwithinthiscapability
and sometimes cannot be done without assistance
externaltothesensor.
NOTE2 The output parameterization is the Star Image
position in the Boresight Reference Frame (BRF),
givenbythetwo
measuresoftheangularrotations
[
]
MEASYMEASXMEAS
sss
,,
,
=
which define the
transformation from the BRF to the star Stellar
ReferenceFrame(SRF).
NOTE3 This capability does not imply to autonomously
identifythestarimagesasimagestobetracked or
explicitly identified by the unit. However, it does
includetheabilitytomaintaintheidentificationof
each
star image and to correctly update the co
ordinates of each image as it moves across the
detectorduetotheangularrateofthesensor.
4.1.4 Autonomous star tracking
4.1.4.1 Inputs
a. The minimum set of inputs to be supplied in order to initialize the
AutonomousStarTrackingshallbe:
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1. theangularrate;
2. thevaliditydate.
b. For aided tracking, data specified in
4.1.4.1ashall be supplied regularly
bythespacecraft,atanupdaterateandaccuracyagreedbythecustomer.
c. Theunitofallinputsshallbeindicated.
4.1.4.2 Outputs
a. A sensor with the autonomous star tracking capability shall have the
minimumoutputs:
1. the position of each star image with respect to a sensordefined
referenceframe;
2. theMeasurementdate.
NOTE This capability does not imply the stars to be
explicitly identified by the unit. However, it
does
includetheabilitytomaintaintheidentificationof
each star image once selected, to correctly update
the coordinates of each image as it moves across
the detector, and autonomouslymanagethe set of
starimagesbeingtracked.
4.1.5 Autonomous attitude determination
4.1.5.1 Inputs
a. Theacquisitioncommandshallbesuppliedasaminimumsetofinputs.
NOTE When a priori initial attitude information for
example an initial quaternion or a restriction
within the celestial sphere, is supplied by the
ground the capability is referred as Assisted
Attitudedetermination
4.1.5.2 Outputs
a. A sensor with autonomous attitude determination shall have the
minimumoutputs:
1. therelativeorientationofthedefinedsensor referenceframewith
respecttothedefinedinertialreferenceframe;
NOTE Therelativeorientationisusuallyexpressedinthe
formofanormalizedattitudequaternion
2. theMeasurementdate;
3.
a validity index or flag estimating the validity of the determined
attitude.
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4.1.6 Autonomous attitude tracking
4.1.6.1 Inputs
a. The minimum set of inputs to be supplied in order to initialize the
AutonomousAttitudeTrackingshallbe:
1. theattitudequaternion;
2. the3dimensionangularrate vector givingthe angularrate of the
sensorBRFwithrespecttotheIRF;
NOTE Thisvectorisexpressedinthe
sensorBRF.
3. thevaliditydateforbothsuppliedattitudeandangularrate.
b. For aided tracking, data specified in
4.1.6.1ashall be supplied regularly
bythespacecraft,atanupdaterateandaccuracyagreedbythecustomer.
c. Exceptforattitudequaternion,theunitofallinputsshallbeindicated.
d. The supplier shall document whether the star sensor initialization uses
either:
Internalinitialization,or
NOTE Theinformationtoinitializethesensorisprovided
by the attitude determination function of the star
sensor.
Directinitialization.
NOTE Theinformationtoinitializethesensorissupplied
byanexternalsourcee.g.AOCS.
4.1.6.2 Outputs
a. A sensor with autonomous attitude tracking capability shall have the
followingminimumoutputs:
1. the orientation of the sensor defined reference frame with respect
tothe inertiallydefinedreferenceframe(nominallyin the form of
anattitudequaternion);
2. theMeasurementdate;
3. a validity index or flag, estimating the
validity of the determined
attitude;
4. measurementofStarMagnitudeforeachtrackedStarImage.
4.1.7 Angular rate measurement
a. A sensor with angular rate measurement capability shall have the
followingminimumoutputs:
1. the instantaneous angular rates
around the Boresight Reference
Frame(BRF)axesrelativetoinertialspace;
2. theMeasurementdate.
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b. The date of measurement shall be expressed as a (scalar) number
indicating the delay) relative to a known external time reference agreed
withthecustomer.
NOTE The intended use of this capability is either when
theattitudecannotbedeterminedortoprovidean
angularrate.
4.1.8 (Partial) image download
4.1.8.1 Image download
a. A sensor with the (partial) image download capability shall have the
followingminimumoutputs:
1. thesignalvalueforeachrelevantdetectorelement;
2. theMeasurementdate.
b. Any use of image compression (e.g. for transmission) shall be
documented.
NOTE The definition of the capability is intended to
exclude
‘lossy’ image compression, though such
compression can be a useful option under certain
circumstances.
4.1.8.2 Image Output Time
a. Thesuppliershallspecifythenumberofbitsperpixelusedtoencodethe
detectorimage.
b. The image output time shall be verified by test using the hardware
agreedbetweenthecustomerandsupplier.
NOTE1 The hardware used to perform the test is the
hardware used to download
the image from the
starsensor.
NOTE2 Forexample:
“TheStarSensorshallbecapableofperforming
a full Image Download of the entire Field of
View at 12bit resolution. The image output
timeshallbelessthan10seconds.”
“TheStarSensorshallbecapable
ofperforming
a partial Image Download at 12bit resolution
ofan×nsectionoftheFieldofView.Theimage
outputtimeshallbelessthan10seconds.”
4.1.9 Sun survivability
a. Asensorwiththesunsurvivabilitycapabilityshallwithstanddirectsun
illuminationalongtheboresightaxis,foratleasta given periodof time
agreedwiththecustomer,withoutsubsequentpermanentdamage.
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b. Asensorwiththesunsurvivabilitycapabilityshallrecoveritsfullquoted
performancesafterthesunaspectanglehasbecomegreaterthanthesun
exclusionangle.
4.2 Types of star sensors
4.2.1 Overview
Thissubclausespecifiesthenomenclatureusedtodescribethedifferenttypesof
starsensors.Theirclassificationisbasedontheminimumcapabilitiestobemet
byeachtype.
The term star sensor is used to refer generically to any sensor using star
measurementstodriveitsoutput.Itdoesnotimply
anyparticularcapabilities.
NOTE The term Star Scanner is used to refer to a Star
Sensoremployedonspinningspacecraft.Thiskind
of sensor performs star measurements at high
angular rate (tens of deg/s). Formal capability
definition of the Star Scanner, together with
definedperformancemetricsareoutsidethescope
ofthisspecification.
4.2.2 Star camera
a. Astarcamerashallincludecartographyasaminimumcapability.
4.2.3 Star tracker
a. Astartrackershallincludethefollowingminimumcapabilities:
1. cartography;
2. startracking.
NOTE If the autonomous star tracking capability is
present, the cartography capability is internal to
the unit when initializing the tracked stars and
hencetransparenttotheground.
4.2.4 Autonomous star tracker
a. An autonomous star tracker shall include the following minimum
capabilities:
1. autonomousattitudedetermination(‘lostinspace’solution);
2. autonomousattitudetracking(withinternalinitialization).
b. The supplier shall document whether the autonomous attitude
determination capability is repetitively used to achieve the autonomous
attitudetracking.
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4.3 Reference frames
4.3.1 Overview
Thestandardreferenceframesaredefinedin3.2.3.
Otherintermediatereferenceframesaredefinedbythemanufacturersinorder
to define specific error contributions, but are not defined here,as they are not
usedintheformulationoftheperformancemetrics.Seealso
AnnexF.
4.3.2 Provisions
a. Any use of an IRF shall be accompanied by the definition of the IRF
frame.
b. Anyuseofanattitudequaternionshallbeaccompaniedbythedefinition
oftheattitudequaternion.
4.4 On-board star catalogue
a. The supplier shall state the process used to populate the onboard star
catalogueandtovalidateit.
b. Theprocessstatedin
4.4ashallbedetailedtoalevelagreedbetweenthe
customerandthesupplier.
c. The supplier and customer shall agree on the epoch at which the on
boardstarcatalogueisvalid.
NOTE In this context, ‘valid’ means that the accuracy of
the onboard catalogue is best (e.g.
the effect of
propermotionandparallaxisminimized).
d. The supplier shall state the epoch range over which performances are
metwiththeonboardstarcatalogue.
e. The supplier shall deliver the onboard star catalogue, including the
spectralresponsesoftheopticalchainanddetector.
f. If the
star sensor has the capability of autonomous attitude
determination, the supplier shall deliver the onboard star pattern
catalogue.
g. The maintenance process of the onboard star catalogue shall be agreed
betweenthecustomerandthesupplier.
NOTE1 Themaintenanceprocessincludesthecorrectionof
parallax and the
correction of the star proper
motionsintheonboardstarcatalogue.
NOTE2 Themaintenanceprocessincludesthecorrectionof
the onboard catalogue errors identified in flight
(e.g.magnitude,coordinates).
h. The supplier shall state any operational limitations in the unit
performancecausedbytheonboardcatalogue(e.g.
autonomousattitude
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determinationnotpossibleforsomeregionsinthesky).Theselimitations
shallbeagreeduponbetweenthesupplierandthecustomer.
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5
Performance requirements
5.1 Use of the statistical ensemble
5.1.1 Overview
Performances have a statistical nature, because they vary with time and from
one realization of a sensor to another. This clause presents the knowledge
requiredtobuildperformancesup.Fulldetailscanbefoundin
AnnexE.
Onlyanenvelopeoftheactualperformancescanbeprovided.Centraltothisis
the concept of a
‘statistical ensemble’, made of ‘statistical sensors (i.e. not
necessarily built, but representative of manufacturing process variations) and
observations(dependingontimeandmeasurementconditions).
Three approaches (calledstatistical interpretations) can be taken to handle the
statistical
ensemble:
Temporalapproach:performancesareestablishedwithrespecttotime.
Ensemble approach: performances are established on statistical
sensors
(i.e.notnecessarilybuilt),attheworstcasetime.
Mixedapproach,whichcombinesboththeapproachesabove.
The conditions elected to populate the statistical
ensemble are defined on a
casebycase basis for each performance parameter, as described in the
followingclauses.
5.1.2 Provisions
a. Theperformancesshallbeassessedbyusingtheworstcasesensorofthe
statistical
ensemble.
b. The statistical ensemble shall be characterized and agreed with the
customer.
c. The performances shall be assessed by using the sensor EOL conditions
agreedwiththecustomer.
NOTE The EOL conditions include e.g. aging effects,
radiationdose.
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5.2 Use of simulations in verification methods
5.2.1 Overview
Simulations efficiently support the verification of performances. A set of
simulationsprovidesan estimateofaperformance,obtainedbyprocessingthe
simulation results in a statistical fashion. Because the set of simulations is
limited, the performance estimated by simulations has a given accuracy,
essentiallydependingonthenumberofsimulations.
5.2.2 Provisions for single star performances
a. Software models of singlestar measurement error shall be validated for
singlestarperformance(atzerobodyrates)againstongroundtestsusing
artificialstellarsources.
NOTE Denoting the confidencelevel to be verified asP
C,
and assuming that the performance confidence
level result to be obtained is to an accuracy ΔP
with95%estimationconfidencelevel,thenumber
of MonteCarlo runs to be performed is greater
than
2
)1(4
P
PP
CC
Δ
.
5.2.3 Provisions for quaternion performances
a. Software models of attitude quaternion error shall be validated against
onground tests using artificial stellar sources or with on ground tests
agreedbythecustomer.
NOTE1 Denoting the confidencelevel to be verified asP
C,
and assuming that the confidence result to be
obtained is to an accuracy ΔP with 95%
confidence,the number ofMonteCarloruns to be
performedisgreaterthan
2
)1(4
P
PP
CC
Δ
.
NOTE2 RefertoAnnex
E.1forfurtherdetails.
5.3 Confidence level
a. The following confidence level shall be agreed with the customer (see
5.5):
1. forthethermoelasticerror;
2. fortheFOVspatialerror;
3. forthepixelspatialerror;
4. forthetemporalnoise;
5. forthemeasurementdateerror;
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6. RefertoAnnexEforfurtherdetails.
NOTE1 A performance confidence level of 95% is
equivalent to a 2 sigma confidence level for a
Gaussiandistribution.
NOTE2 A performance confidence level of 99,7% used is
equivalent to a 3 sigma confidence level for a
Gaussiandistribution.
5.4 General performance conditions
a. The performance conditionsof the ‘statistical ensemble’shallbe used to
encompassthefollowingconditionsforEOL:
1. worstcasebaseplatetemperaturewithinspecifiedrange;
2. worstcaseradiationfluxwithinspecifiedrange;
3. worstcase stray light from solar, lunar, Earth, planetary or other
sources.
NOTE1 InadditionvaluesforBOLcanbegiven.
NOTE2 Worstcasestraylight
conditionsarewiththeSun,
Earth and (where appropriate) Moon
simultaneously at their exclusion angles together
with worstcase conditions for any other light
sources.
b. Themaximummagnitudeofbodyrateshallbeused.
NOTE The maximum body rate is the worst case
conditionformostmissions.Forspecific
cases,the
worstcasecanbeadapted,e.g.toincludejitter.
c. The supplier shall identifytheworst case projection in BRF of the value
definedin
5.4d.
NOTE Different angular rates can be specified with
associatedrequiredperformance.
d. Themaximummagnitudeofangularaccelerationshallbeused.
NOTE The maximum angular acceleration is the worst
case condition for most missions. For specific
cases, the worst case can be adapted, e.g. to
includejitter.
e. The
supplier shall identifytheworstcase projection inBRF of thevalue
definedinbullet
d.
NOTE Different angular accelerations can be specified
withassociatedrequiredperformance.
f. Formultipleheadconfigurationtheworstcaseconditionsofangularrate
and stray light of each optical head shall be discussed and agreed
betweensupplierandcustomer.
g. Single star position measurement performance within the verification
simulations
shallbe:
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1. validated against onground test data for fixed pointing
conditions,and
2. abletopredictmetricperformanceundertheseconditionswithan
accuracyof10%.
h. If test data is available for the individual error sources, the simulation
shallbevalidatedagainstthisdatawithanaccuracyof10%.

i. Detector error sources in the simulation shall also be validated using
directdatainjectionintotheelectronicsandanalysisofthetestoutputs.
j. The simulation allows the verification to cover the full range of
conditions, including stray light, finite rates/accelerations, full range of
instrumentmagnitudes,andtheworst
caseradiationexposure.
k. EOL simulations used to predict EOL performance shall be verified by
testcasesverifiableagainstmeasurableBOLdata.
l. Theimpactofindividualstarerrorsonthe overallrateaccuracyshallbe
providedviasimulation.
m. Noaidedtrackingshallbeconsidered.
5.5 General performance metrics
5.5.1 Overview
Clause 5.5 presents the general performance metrics for the error contributing
tothestarsensorperformances.In
AnnexH,anexampleofdatasheetbuilton
theperformancemetricsisgiven.
5.5.2 Bias
5.5.2.1 General
a. Theconfidencelevelspecifiedinclause5.3shallbeused.
b. The‘Ensemble’interpretationshallbeusedasfollows:
NOTE TheEnsembleinterpretationisasfollows:
A
statistical collection of sensors is arbitrarily
chosen.
A given set of observations is arbitrarily
chosen.
The specification for this type of variability is
‘lessthanthelevelSinconfidenceleveln%ofa
statistical ensemble of sensors/observations for
theworstcasetime’.
c. The bias performance shall be specified for a defined ambient
temperature.
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NOTE The initial alignment is an instantaneous
measurement error at the time of calibration. For
the purposes of error budgeting it can be
consideredtobeaninvarianterror.
5.5.2.2 Contributing error sources
a. Thefollowingtypesoferrorsourceshallbeincluded:
1. Onground calibration error between the sensor Alignment
ReferenceFrame(ARF)andthe sensorBoresightReferenceFrame
(BRF).
NOTE This arises typically from accuracy limitations
within the measurement apparatus used to
performthecalibration.
2. Launchinducedmisalignmentsof
BRFwithrespecttoMRF.
3. Spatialerrorincaseofinertialpointing.
NOTE Refer to the
Annex G for the contributing error
sourcesdescription.
5.5.2.3 Verification methods
a. Thecalibrationshallbeperformedviagroundbasedtestusinganoptical
bench setup to determine the sensor Alignment Reference Frame
(ARF)‐sensorBoresightReferenceFrame(BRF)alignment.
b. The bias error shall be validated by analysis, test or simulation, taking
intoaccountcalibrationtestbenchaccuracy.
NOTE1
Initial alignment verification cannot be done
without verification of the measurement accuracy
ofthesetupusedforcalibration.
NOTE2 E.g. “The Star Sensor initial alignment shall have
an initial alignment error (X, Yaxes rotation) of
less than 10 arcsec at a quoted ambient
temperature(thetemperature
duringalignment).”
5.5.3 Thermo elastic error
5.5.3.1 General
a. Theconfidencelevelspecifiedinclause5.3shallbeused.
b. The‘Ensemble’interpretationshallbeused(seeNOTEin
5.5.2.1b).
NOTE The ‘Ensemble’ interpretation is selected here as
thetimevariationoftheseerrorsisslowtheyare
to all intents and purposes biases for practical
measurementscenarios.
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5.5.3.2 Contributing error sources
a. Error sources that gradually change the alignment of the sensor
Mechanical Reference Frame (MRF) and the sensor Boresight Reference
Frame(BRF)fromthestartoftheinflightmissionshallbeincluded.
NOTE E.g.“Thethermalsensitivitytotemperatureofline
ofsightstabilityshallbelessthan1
arcsec/Kelvin.”
5.5.3.3 Verification methods
a. Thermallyinducederrorcontributionstothethermoelasticerrorshallbe
verified by the use of thermal models supported and validated by
groundtestresultsperformedunderthermalvacuumconditions.
5.5.4 FOV spatial error
5.5.4.1 General
a. Theconfidencelevelspecifiedinclause5.3shallbeused.
b. The‘Ensemble’interpretationshallbeused(seeNOTEin
5.5.2.1b).
c. The performance shall be specified under the related performance
generalconditions.
5.5.4.2 Contributing error sources
a. ContributingErrorSourcesshallinclude:
1. pointspreadfunctionvariabilityacrosstheFOV;
2. residual of calibration of focal length (including its temperature
sensibility)andopticaldistortions(includingchromatism);
3. residual of aberration of light in case where it is corrected at
quaternionlevelandnotatstarlevel;
4.
CCD,CTEeffect(includingitsdegradationsduetoradiations);
5. catalogueerror(includingstarpropermotionandparallax).
5.5.4.3 Verification methods
a. ThemeasurementoftheFOVspatialerrorshallbeperformedviaground
test (for contributing error sources
5.5.4.2a.1 and 5.5.4.2a.2) and by
analysis (for contributing error sources
5.5.4.2a.3, 5.5.4.2a.4 and
5.5.4.2a.5).
b. Radiationeffectsshallbesupportedbytestresults.
NOTE E.g. “The Star Sensor shall have a FOV spatial
errorlessthan10arcsecon X,Yaxesand40arcsec
on Z axis for spatial period smaller than
5degrees.”
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5.5.5 Pixel spatial error
5.5.5.1 General
a. Theconfidencelevelspecifiedinclause5.3shallbeused.
b. The‘Ensemble’interpretationshallbeused.
c. The performance shall be specified under the related performance
generalconditions.
5.5.5.2 Contributing error sources
a. Contributingerrorsourcesshallconsistofatleast:
1. detectorPhotoResponseNonUniformity(PRNU);
2. detectorDarkSignalNonUniformity(DSNU);
3. detectordarkcurrentspikes‐ifrelevantaccording tothedetector
technology;
4. detector Fixed Pattern Noise (FPN)‐if relevant according to the
detectortechnology;
5. starcentroid
computationerror(interpolationerror).
b. Allothererrorsourceswithrelevantspatialbehaviourshallbeidentified
bythesupplierandusedfortheassessmentofperformances.
5.5.5.3 Verification methods
a. Contributingerrorsourcesshallbeverifiedbyongroundtests.
b. Pixel spatial errors shall be verified by analysis and simulations using
verifiedbudgetsofcontributingerrorsourcesmethods.
NOTE E.g. “The Star Sensor shall have a pixel spatial
errorof less than 5arcseconds(resp.30)aroundX
and Y axes (resp. Z axis) for spatial period of 400
arcsecond, and less than 2 arcseconds (resp 10)
around X and Y axes (resp. Z axis) for spatial
periodof100arcsecond.”
5.5.6 Temporal noise
5.5.6.1 General
a. Theconfidencelevelspecifiedinclause5.3shallbeused.
b. The‘temporal’interpretationshallbeused,andtheperformanceshallbe
specifiedundertherelatedperformancegeneralconditions.
5.5.6.2 Contributing error sources
a. TheContributingErrorSourcesshallinclude:
1. shotnoiseonstarsignal;
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2. shotnoiseonbackgroundsignalanddarkcurrent;
3. readoutnoise;
4. quantificationnoise;
5. datationnoise.
NOTE1 Temporal noise depends on exposure time and
detectortemperatures.
NOTE2 Noise contributors at star level depend on star
magnitude angular rates/acceleration, and
optics/detector characteristics (e.g. exposure time,
opticalcontamination,transmissionloss,defocus).
NOTE3 Datation noise is the temporal noise part of the
measurementdateerrordescribedin
5.5.8.
5.5.6.3 Verification Methods
a. Temporalnoiseshallbeestimatedbysimulation.
b. Errorsourcecontributors
5.5.6.2a.1,5.5.6.2a.2,5.5.6.2a.3,5.5.6.2a.4shallbe
validatedagainstongroundtestdataat(BOL)forfinitescenarios.
c. Error source contributor
5.5.6.2a.5 (datation noise) shall be assessed by
analysis.
NOTE1 Night Sky tests are not used as single verification
methodduetoexperimentalconditions.Nightsky
tests can be used to assess temporal noise in
addition to other required verification methods
(simulationsandongroundtests).
NOTE2 E.g.“The Star
Sensor shall have temporal noise of
less than 10arcsecaround any axisup to 10 deg/s
atEOLandforaccelerationsupto1,0deg/s².”
5.5.7 Aberration of light
5.5.7.1 General
a. The supplier shall document what type of relativistic correction is
performed.
b. The supplier shall document the maximum error and minimum
frequencyofthespacecraftvelocityprovidedtothesensor.
5.5.7.2 Contributing error sources
a. ThecontributingErrorSourcesshallinclude:
1. Absolutelinearvelocityofthespacecraftwithrespecttothesun.
2. Accuracy of the velocity information (or propagation) used for
correction.
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5.5.7.3 Verification methods
a. The correction software shall be validated, comparing the computed
correctiontermwiththeanalyticalexpression.
NOTE1 Thiserrorcorrectionisdifficulttoverifysinceitis
atheoreticaltermoferror.
NOTE2 E.g. “The relativistic effect has an impact of less
than 0,07’’ (3σ) at quaternion level.
The needed
accuracyofthevelocityofthespacecraftdelivered
to the star sensor shall be better that 100m/s, at a
frequencyof0,1Hz.”
5.5.8 Measurement date error
a. Theconfidencelevelspecifiedinclause5.3shallbeused.
b. TheMeasurementdateErrorshallbeverifiedbytest.
NOTE E.g. “The Measurement date Error shall be less
than0,1ms.”
5.5.9 Measured output bandwidth
a. The bandwidth shall be verified by analysis of the Integration Time,
output
Sampling Time and any onboard data filtering that can be
present.
b. Ongroundtestsmaybeperformed.
NOTE E.g. “The
Star Sensor shall have a Measured
OutputBandwidthofgreaterthan10Hz.”
5.6 Cartography
a. For star position measurements, the performance conditions of the
‘statistical
ensemble’shallbeusedtoencompassthefollowingconditions
forBOL:
1. worstcasestarlocationinFOV;
2. worstcaseStarMagnitudewithinspecifiedrange.
5.7 Star tracking
5.7.1 Additional performance conditions
a. For star position measurements, the performance conditions of the
‘statistical
ensemble’shallbeusedtoencompassthefollowingconditions
forBOL:
1. worstcasestarlocationinFOV;
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2. worstcaseStarMagnitudewithinspecifiedrange.
5.7.2 Single star tracking maintenance probability
a. Thefollowingconditionsshallbemet:
1. quote the maximum body rate ω
CROSS, MAX around the sensor
Boresight Reference Frame (BRF) X‐ or Yaxes and ωZ, MAX
around the BRF Zaxis for which the single star tracking
maintenanceprobabilityisachievedoverthedefinedtimeperiod;
2. quote the maximum body angular acceleration around the sensor
boresight reference frame (BRF) X‐ or
Y‐ axes and the maximum
body angular acceleration around the BRF Zaxis for which the
single star tracking maintenance probability is achieved over the
definedtimeperiod.
NOTE E.g. “The Track Maintenance Probability shall be
greater than 99% over a time period of 1 minute
for a tracked
Star Image (of magnitude less than
tbdmi)remainingwithinthesensorFOV,forrates
aroundanyaxisofupto100arcsec/satEOL,with
accelerationsupto10arcsec/s².”
5.8 Autonomous star tracking
5.8.1 Additional performance conditions
a. For star position measurements, the performance conditions of the
‘statistical
ensemble’shallbeusedtoencompassthefollowingconditions
forBOL:
1. worstcasestarlocationinFOV;
2. worstcaseStarMagnitudewithinspecifiedrange.
b. The followingadditionalperformance metrics shallbe established:track
maintenanceprobability,asspecifiedin
5.7.2.
c. Forthestatistical
ensemble,provisionsin5.2.2shallbeapplied.
NOTE The same definition for the ‘statistical ensemble’
givenin5.1.1applies.
5.8.2 Multiple star tracking maintenance level
a. Thefollowingconditionsshallbemet:
1. quote the maximum body rate ω
CROSS, MAX around the sensor
Boresight Reference Frame (BRF) X‐ or Yaxes and ωZ, MAX
around the BRF Zaxis for which the multiple star tracking
maintenancelevelisachievedoverthedefinedtimeperiod;
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2. quote the maximum body angular acceleration around the sensor
boresight reference frame (BRF) X‐ or Y‐ axes and the maximum
body angular acceleration around the BRF Zaxis for which the
single star tracking maintenance probability is achieved over the
definedtimeperiod;
3. Thegeneralprovisionsin
5.2.2shallbeapplied.
NOTE E.g.“TheMaintenanceLevelofStarTracksshallbe
atleast5tra cksforatotaltimeof995swithinany
1000s period, for rates around any axis of up to
100arcsec/s at EOL, andforaccelerationsupto10
arcsec/s².”
5.9 Autonomous attitude determination
5.9.1 General
a. When Autonomous Attitude Tracking is performed by using repetitive
AutonomousAttitudeDeterminationthemetricsrelativetoautonomous
attitudetrackingspecifiedin
5.10shallbeapplied.
NOTE Thiscapabilityis often referred toastheabilityto
solve the ‘lost in space’ problem. The orientation,
orattitude, measurement is nominally in the form
of a quaternion that parameterizes the
transformation between the Inertial reference
frameandthesensordefinedreferenceframe.The
determination is nominally performed by
comparing star images measured on a detector to
knownstarpositionsandcharacteristicsstoredina
starcataloguewithinthesensor.
b. When Autonomousattitude determinationisonly used for autonomous
attitude tracking initializationthe general performance metrics shall not
beused.
5.9.2 Additional performance conditions
5.9.2.1 Autonomous attitude determination
a. The Autonomous attitude determination shall be subjected to the
followingattitudedeterminationprobabilityperformancemetrics:
1. probabilityofcorrectattitudedetermination;
2. probabilityoffalseattitudedetermination;
3. probabilityofinvalidattitudedetermination.
NOTE Thevalidityflagneedsnotaperformancemetric.
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5.9.2.2 Lunar and planetary effects on performance
a. If a statement of operation with the Moon in the FOV is specified, the
attitude determination probabilities shall be quoted for the ‘Moon in
FOV’scenario.
b. IfastatementofoperationwithplanetaryobjectsintheFOVisspecified,
theattitudedeterminationprobabilitiesshallbequotedforthe‘Planet
in
FOV’scenario.
c. The attitude determination probabilities specification shall be quoted
with the maximum number of False Stars in the FOV for which the
specificationissatisfied.
5.9.3 Verification methods
a. Theprobabilitiesofattitude determination specificationshallbeverified
byapplyingthegeneralprovisionsin
5.2.2and5.2.3.
b. Functionalverificationmaybeperformedbymeansofanightskytest.
5.9.4 Attitude determination probability
a. ProbabilityofCorrectAttitudeDetermination:
1. Thecorrectattitudethresholdshallbespecified.
NOTE E.g. “The correct attitude threshold shall be
0,1degree around X an Y axis and 0,3 degree
aroundZaxis
2. Theprobabilityofcorrectattitudedeterminationshallbeestimated
consideringallpossibleinitialpointing
directionswithinadefined
regionwithinthecelestialsphere.
3. Theprobabilityofcorrectattitudedeterminationshallbeestimated
undertheconditionsgivenin
5.4and5.9.2.
4. The probability of correct attitude determination shall be verified
usingthemethodspecifiedin
5.9.3.
NOTE E.g. “An example of requirement specification is
the following: the probability of correct attitude
determination within 10s shall be greater than
99,99% for random initial pointings within the
entirecelestialsphere,forratesaroundanyaxisof
upto 100arcsec/s at EOL and for accelerations
up
to10arcsec/s².”
b. ProbabilityofFalseAttitudeDetermination:
1. The probability of false attitude determination shall be estimated
consideringallpossibleinitialpointingdirectionswithinadefined
regionwithinthecelestialsphere.
2. The probability of false attitude determination shall be estimated
undertheconditionsgivenin
5.4and5.9.2.
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3. The probability of false attitude determination shall be verified
usingthemethodspecifiedin
5.9.3.
NOTE E.g. “The probability of false attitude
determination within 10s shall be less than 0,1%
for random initial pointings within the entire
celestialsphere, for ratesaroundanyaxisofup to
100arcsec/s at EOL and for accelerations up to
10arcsec/s².”
c. ProbabilityofInvalid
AttitudeSolution:
1. The probability of invalid attitude solution shall be estimated
consideringallpossibleinitialpointingdirectionswithinadefined
regionwithinthecelestialsphere.
2. The probability of invalid attitude determination shall be
estimatedundertheconditionsgiveningivenin
5.4and5.9.2.
3. The probability of invalid attitude determination shall be verified
usingthemethodspecifiedin
5.9.3.
NOTE E.g. “The probability of invalid attitude solution
shallbelessthan0,1%forrandominitialpointing
within the entire celestial sphere, for rates around
any axis of up to 100arcsec/s at EOL and for
accelerationsupto10arcsec/s².”
5.10 Autonomous attitude tracking
5.10.1 Additional performance conditions
a. ForbothBOLandEOL,theperformancemetricsshallbespecifiedeither:
Fromthewholecelestialsphereincludingthevaultinthestatistics,
or
NOTE The statistical
ensemble is then composed of
measurements randomly performed on the entire
celestialvault.
Fromasetoffixeddirectionsinthecelestialsphere.
b. If the metrics are specified from a set of fixed directions in the celestial
sphere when satisfying conditions detailed in
a the following shall be
met:
1. assess the metrics for each direction, limiting the statistical
ensembletomeasurementsperformedinthisdirectiontocompute
theperformance;
2. Specifyallorpartofthefollowing:
(a) Themeanperformanceamongallperformancesachievedin
thedirectionsofthecelestialsphere,
(b) Thevalueachievedonn%ofthecelestialsphere,
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NOTE This is the performance achieved for n% of the
pointing directions within the whole celestial
vault. If n is not quoted, a value of 99% is
assumed.
(c) The value achieved in the worstcase direction of the
celestialsphere.
NOTE Thisdirectionisrelatedtotheworstdistribution
of
starsoverthestarsensorFieldofView,takinginto
accountembeddedalgorithmsandcatalogues.The
statistical
ensemble is then reduced to
measurementsperformedinthisdirection.
c. Performances may also be specified for a restricted area of the celestial
sphereagreedwiththecustomer,inwhichcasetheperformancemetrics
arethenspecifiedinthesameway,limitingthestatistical
ensembletothe
specifiedarea.
d. ForLunarandplanetaryeffectsonperformancethefollowingconditions
shallbemet:
1. IfastatementofoperationwiththeMoonintheFOV isspecified,
quote the probability ofmaintenance of tracking for the ‘Moon in
FOV’scenario.
2. If a statement of operation with planetary
objects in the FOV is
specified, quote the probability of maintenance of tracking the
‘PlanetinFOV’scenario.
e. FortheeffectofFalseStarsthefollowingconditionshallbemet:
Quote the maintenance level of tracking with the maximum number of
FalseStarsintheFOVforwhichthe
specificationisapplicable.
f. Forthe effectofsingleeventupsets(SET’s)thefollowingconditionshall
bemet:
Quote the maintenance level of tracking with the maximum number of
SET’spersecondforwhichthespecificationisapplicable.
5.10.2 Maintenance level of attitude tracking
5.10.2.1 General
a. The performance shall be specified under the conditions given in 5.10.1
and
5.10.2.2a.
5.10.2.2 Verification methods
a. The maintenance level of tracking shall be verified by applying the
generalprovisionsin
5.2.2.
NOTE E.g. “The maintenance level of tracking shall be
more than 995s within a 1000s period, for rates
around any axis of up to 100arcsec/s at EOL, and
foraccelerationsupto10arcsec/s².”
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5.10.3 Sensor settling time
a. Theperformanceshallbespecifiedundertheconditionsgivenin5.10.1.
b. Forlunarandplanetaryeffectson performancethe following conditions
shallbemet:
1. IfastatementofoperationwiththeMoonintheFOV isspecified,
quotetheSensorSettlingTimeforthe‘MooninFOV’scenario.
2. If a statement of operation with planetary objects in the
FOV is
specified, quote the Sensor Settling Time for the ‘Planet in FOV’
scenario.
c. For the effect of False Stars the following condition shall be met: Quote
theSensorSettlingTimewiththemaximumnumberofFalseStarsinthe
FOVforwhichthespecificationisapplied.
NOTE The
effect of convergence of internal algorithm
shallbeconsidered
d. The Sensor Settling Time shall be verified by applying the general
provisionsin
5.2.2.
NOTE E.g.“SensorSettlingTime shallbe less than 5s for
more than 99% of random initial pointing within
the entire celestial sphere, for rates around any
axis of up to 100arcsec/s at EOL and for
accelerationsupto10arcsec/s².”
5.11 Angular rate measurement
5.11.1 Additional performance conditions
a. Additionalperformanceconditions,definedin5.10.1shallbeapplied.
b. Contributingerrorsourcesshallbeestablished.
NOTE They are a function of the precise technique used
todeterminetherate.
5.11.2 Verification methods
a. Performanceat finiteratesandaccelerations,andforall scenariosunder
thespecifiedconditions,shallbeverifiedbysimulation.
NOTE E.g. The Star Sensor shall have an angular rate
measurement around any BRF axis of less than
100arcsec/s, at rates around any axis of up to
10deg/s
at EOL and for accelerations up to
1deg/s².
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5.12 Mathematical model
a. The supplier shall deliver a temporal functional mathematical model of
theperformanceofthestarsensor.
NOTE This is essential for some capabilities
(e.g.autonomousattitudetracking).
b. The functional mathematicalmodelshall be representative of the sensor
actualtemporalperformancesforrealistickinematicprofiles.
c. The functional mathematical
model shall include environmental
parameters.
d. The functional mathematical model shall be established with customer
approvedmethods.
e. The functional mathematicalmodel shallbe validated againstthe actual
temporalperformancesofthesensor.
f. Thesuppliershalldeliver:
eithertheFMMsoftwareusedbythesuppliertoassessthesensor
performancesanditsassociateddocumentation(e.g.usermanual)
inaformatagreedwiththecustomer,or
the FMM DRD ofthesensormodelused by thesupplierto assess
thesensorperformances,inconformancewith
AnnexA.
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Annex A (normative)
Functional mathematical model (FMM)
description - DRD
A.1 DRD identification
A.1.1 Requirement identification and source
document
ThisDRDiscalledfromECSSEST6020,requirement5.12f.
A.1.2 Purpose and objective
The functional mathematical models are established to serve as input for
detailedAOCSanalysesanddetailedperformancesimulations.
A.2 Expected response
A.2.1 Scope and content
<1> Introduction
a. The FMM description shall contain a description of the purpose,
objective,contentandthereasonpromptingitspreparation.
b. Any open issue, assumption and constraint relevant to this document
shallbestatedanddescribed.
c. Statusandlimitationsofthemodelshallbedescribedindetail.
<2> Applicableandreferencedocuments
a. TheFMMdescriptionshalllisttheapplicableandreferencedocumentsin
supporttothegenerationofthedocument.
<3> Definitionsandabbreviations
a. The FMM description shall list the applicable directory or glossary and
themeaningofspecifictermsorabbreviationsutilizedintheFMM.
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<4> Functionalmathematicalmodel(FMM)
a. The steps from the actual quaternion in inertial frame to the sensor
outputsshallbedocumented,including:
1. staridentification;
2. patternrecognition;
3. star corrections (e.g. optical aberration correction, relativistic
aberrationcorrection);
4. quaternioncomputation;
5. filtering.
b. TheoutputsoftheFMMshallinclude:
1. themeasuredquaternion
andtimedeliveredbythesensor;
2. thestarmeasurementsandtimesdeliveredbythesensor;
3. thestaridentificationinformation.
c. TheoutputsoftheFMMshall includethe outputs of the sensor detailed
inclause
4(see4.1.2.2,4.1.3.2,4.1.4.2,4.1.5.2,4.1.6.2,or4.1.8.2),according
tothesensorcapabilities.
d. TheparametersoftheFMMshallbedocumented.
e. Modelling constraints and critical implementation issues shall be
describedandtheirrelevanceonperformanceshallbeindicated.
f. The FMM shall present the expected temporal outputs of the sensor
modelforgiveninputprofiles.
<5> Modes
a. For sensors with the autonomous attitude determination capability, the
FMM description shall include the autonomous attitude determination
capability.
b. For sensors with the autonomous attitude tracking capability, the FMM
descriptionshallincludetheautonomousattitudetrackingcapability.
<6> Softwaretools
a. The software tools to be used for development of the FMM shall be
specified.
<7> Filesandlists
a. Thefollowinginformationshallbeattachedtothedocument:
1. identificationofdeliveredcomputerfiles;
2. FMMsourcelistsbasedonappliedtools.
A.2.2 Special remarks
None.
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Annex B (informative)
Ancillary terms in Star Sensors
B.1 Overview
This annex standardizes the meaning of terms that, although not used in this
document, are used in star sensors engineering. It also presents the
measurementerrormetrics.
B.2 Time and frequency
B.2.1 frame frequency
inverseoftheframetime
B.2.2 frame time
time interval between two consecutive beginnings of integration time of each
outputofasingleOpticalHead
B.2.3 internal sampling time
time interval between the Measurement Dates of consecutive measurements
fromasingleOpticalHead
B.2.4 internal sampling frequency
inverseoftheinternalSamplingTime
B.2.5 latency
timebetweenthemeasurementdateandtheoutputdate
B.2.6 output date
dateofthefirstavailabilityoftheoutputdataforuseexternaltothesensor
NOTE Sensors can either be operated asynchronously
(output provided when available based on sensor
clock)orsynchronously(whenthesensorisaslave
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to an external clock pulse). In the latter case the
output data sometimes cannot be accessed and
placed in TM until some time after it was made
available. This additional delay is specifically
excludedfromthelatencydefinition.
B.2.7 output rate
rate at which the sensor delivers its data for each output of a single Optical
Head
B.3 Angles of celestial bodies
B.3.1 acquisition angle with Moon angle (AAM)
lowest Aspect Angle of the Full Moon at which the Autonomous Attitude
Determinationisoperatingsuccessfullybutwithdegradedperformance
NOTE1 AAMislessorequaltoMEAandisexpectedtobe
greaterorequaltoTAM.
NOTE2 AAM and TAM define the robustness of the
behaviourof
thestarsensorwhentheMoonenters
thefieldofview.
B.3.2 tracking angle with moon in the FOV (TAM)
lowest Aspect Angle of the Full Moon at which the Autonomous Attitude
Trackingisstilloperatingsuccessfullybutwithdegradedperformance
NOTE1 TAMislessorequaltoMEA.
NOTE2 TAMandAAM(see
B.3.1)definetherobustnessof
the behaviour of the star sensor when the Moon
entersthefieldofview.
B.4 Full sky
celestial sphere covering the complete 4π steradian solid angle with respect to
thesensor
B.5 Measurement error metrics
B.5.1 Overview
Thisclause declinesthe measurement error metrics,prior to application to the
Star Sensor measurement error specification. A link to the nomenclature for
traditionalerrormetricsisalsoincludedtoaidmigrationtothenewmetricset.
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Annex F establishes the expression of the angular error on which the angular
metricsisapplied:
()
Δ
Δ
Δ
=
ψ
θ
φ
ε
t
B.5.2 time interval for a metric
thetimeintervaltXforametricXisdefinedasatimeperiodwithstarttimetSX
andlengthτ
X
B.5.3 absolute measurement error (AME)
theabsolutemeasurementerror(AME(t))istheangularerror
()
t
ε
atatimet:
()
ttAME
ε
=)(
NOTE Thisisillustratedschematicallyin
FigureB1fora
singleaxisrotationcase.
B.5.4 quaternion absolute measurement error (AMEq)
AMEinwhichtheangularerrorisderivedfromthemeasuredquaternion
meas
q
NOTE The quaternion
meas
q is used to build the frame
transform matrix
M
XRFIRF
T
from an inertial
referenceframe(IRF)toasensordefinedreference
frame (see clause
3.2.3), generically called XRF.
The error
()
Δ
Δ
Δ
=
ψ
θ
φ
ε
t
is then computed from
M
XRFIRF
T
accordingtoAnnexF.
B.5.5 star absolute measurement error (AMEs)
AME in which the angular error is derived from the sensormeasured star
position
][
,, MEASYMEASXMEAS
sss =
NOTE1 The sensormeasured star position
][
,, MEASYMEASXMEAS
sss
=
is defined as two
angular rotations parameterizing the
transformationbetweenasensordefinedreference
frame(heredenoted genericallyby ‘XRF’) and the
Stellar Reference Frame defined by the observed
star for the specific star. The X and Y rotations
provides the full parameterization since the third
rotation is zero by
definition. In this case, the star
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position measurement
MEAS
s
is used to build the
frame transform matrix
M
XRFIRF
T
, from which the
error
()
Δ
Δ
Δ
=
ψ
θ
φ
ε
t
canbecomputedaccordingto
AnnexF.
NOTE2 Theusualparameterizationistousethe‘2’and‘1
Euler rotations (within the 321 convention‐the
anglesaresmallandsotheorderoftherotationsis
not important). Note that, in this definition, these
rotationerrorsare(inthesmallanglelimit)around
the X‐ and Yaxes of the Stellar Reference Frame
(SRF), which are perpendicular to the LOS to the
starinthefieldofview.
B.5.6 absolute rate measurement error (ARME)
the difference between the measured and actual angular rate components,
relativetoitstargetframe,definedas:
()
BRF
M
BRF
tARME
ωω
=
where
M
BRF
ω
and
BRF
ω
are respectively the measured and actual angular rate
vectoraroundtheBoresightReferenceFrameaxes,relativetoinertialspace.
NOTE TheAbsoluteRateMeasurementErrorisspecified
for each axis by the absolute valueof the relevant
vectorcomponent.
B.5.7 mean measurement error (MME)
themeanvalue
ε
oftheangularerror
(
)
t
ε
overatimeinterval
τ
:
()
ε
=ΔtMME where
()
+
=
τ
ε
τ
ε
t
t
dtt
1
NOTE Thisisillustratedschematicallyin
FigureB1fora
singleaxisrotationcase.
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M
ME
ε
t
τ
)(tAME
FigureB1:AME,MMEschematicdefinition
B.5.8 quaternion mean measurement error (MMEq)
MME in which the angular error is derived from the measured quaternion
meas
q .
NOTE Seenotein
B.5.4.
B.5.9 star mean measurement error (MMEs)
MME in which the angular error is derived from the measured star position
MEAS
s
NOTE Seenotein
B.5.5.
B.5.10 relative measurement error (RME)
therelativemeasurementError(RME(t))isdefinedasfollows:
()
ε
ε
τ
=,tRME
NOTE Thisisillustratedschematicallyinforasingleaxis
rotationcasein
FigureB2.
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t
R
ME
ε
τ
FigureB2:RMESchematicDefinition
B.5.11 quaternion relative measurement error (RMEq)
RMEinwhichtheangularerrorisderivedfromthemeasuredquaternion
meas
q
NOTE Seenotein
B.5.4.
B.5.12 star relative measurement error (RMEs)
RME in which the angular error is derived from the measured star position
MEAS
s
NOTE Seenotein
B.5.5.
B.5.13 measurement drift error (MDE)
themeasurementdrifterror(MDE(t))isthedifferencebetween two successive
meanmeasurementerrors,separatedby
MDE
t
Δ
asfollows:
() ()
++Δ+
Δ+
=
ττ
ε
τ
ε
τ
t
t
tt
tt
dttdttMDE
MDE
MDE
11
where the lengths of the two successiveintervals are set to identical values
τ
;
bothintervalsarecontainedinalongerintervalwithlength
OBS
τ
NOTE Thisisillustratedschematicallyin
FigureB3fora
singleaxisrotationcase.
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t
τ
MDE
ε
τ
OBS
τ
FigureB3:MDESchematicDefinition
B.5.14 quaternion measurement drift error (MDEq)
MDE in which the angular error is derived from the measured quaternion
meas
q .
NOTE Seenotein
B.5.4.
B.5.15 star measurement drift error (MDEs)
MDE in which the angular error is derived from the measured star position
MEAS
s
NOTE Seenotein
B.5.5.
B.5.16 rotational error
Each of the metrics defined in clause B.5 is parameterized by 3 rotations
j
ε
around axis j of a specific frame F. With respect to this frame, the rotational
errorR
jaroundeachaxisjoftheFframeisgivenby:
jj
R
ε
=
NOTE1 Therotationalerrorisillustratedin
FigureB4.
NOTE2 The applicability of the specification formulation
intermsofdirectionalandrotationalerrorsallows
separate specificationandperformancestatements
relative to the direction of the sensor LOS and
around the sensor LOS. This is useful since the
performance in these 2 areas is typically
significantly different
for single optical head
configurationand hencerequiresseparate
specification.
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B.5.17 directional error
The directional error D(j) for axis j is defined as the halfcone angle between
the measuredand reference position of axis j and is given (for small rotation
angles)by:
22
lkj
D
εε
+=
wherekandlarethetwoaxesperpendiculartoaxisj’.
NOTE1 Thedirectionalerrorisillustratedin
FigureB4.
NOTE2 The applicability of the specification formulation
intermsofdirectionalandrotationalerrorsallows
separate specificationandperformancestatements
relative to the direction of the sensor LOS and
around the sensor LOS. This is useful since the
performance in these 2 areas is typically
significantly different
for single optical head
configurationand hencerequiresseparate
specification.
Axis ‘j
Axis ‘k
Axis ‘l
ε
k
ε
l
D
j
FigureB4:RotationalanddirectionalErrorGeometry
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B.6 Spatial errors
Some error contributors vary with theposition ofastar on the detector.These
errors(e.g.fieldofviewerrors,pixelerrors)canbetackledbyspatialerrors.
Themathematicalexpressionsarethesameastheonespresentedintheclauses
above,inwhichthetimetisessentiallyreplacedby
aspatialpositionx.
Foramoregeneraldomainvariable,x,theindicescanberedefinedasfollows:
() ()
xxAME
ε
=
() ()
Ω
Ω
=
x
xdxxMME
x
ε
1
()() ()
Ω
Ω
=Ω
x
xdxxxRME
x
x
''
1
,
εε

x
x Ω
()
() ()
ΩΩ
Ω
Ω
=ΩΩ
1,21
''
1
''
1
,
1,2,
2,1,
xx
xdxxdxMDE
xx
xx
εε

Where
x
Ω
is a specified region of parameter space and
x
x Ω means that x
lieswithinthatregion.
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Annex C (informative)
Optional features of star sensors
C.1 Overview
Thisannexdefinesoptionalfeaturesorcapabilitiesofstarsensors.Itfollowsthe
same structure as the clause
4 to allow for a direct link between requirements
andoptions.
C.2 Cartography
A sensor with cartography capability can have the following additional
outputs:measurementofstarmagnitudeofeachdetectedstarimage.
NOTE The star images obtained need not be captured at
thesameinstantintime.
C.3 Star tracking
a. Thefollowingadditionalinputstolaunchtrackingcanbeprovided:
1. theangularaccelerationandjerkofthesensorBRFwithrespectto
theIRF,withtheirvaliditydates;
2. theaccuracyofsuppliedinputs.
NOTE1 Angular acceleration and jerk are supplied in the
form of 3dimension vectors
giving the angular
acceleration and jerk of the sensor BRF with
respect to the IRF. These vectors are expressed in
thesensorBRF.
NOTE2 In the case of external inputs coming from the
spacecraftthe starsensorsupplier canindicate the
minimum required accuracy for supplied data in
order
toproperlyswitchintotracking.
b. A sensor with the star tracking capability can have the following
additionaloutputs:measurementofstarmagnitudeforeachtrackedstar
image.
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C.4 Autonomous star tracking
a. Thefollowingadditionalinputstolaunchtrackingcanbeprovided:
1. TheangularaccelerationandjerkofthesensorBRFwithrespectto
theIRF,withtheirvaliditydates.
2. Theaccuracyofsuppliedinputs.
NOTE1 Angular acceleration and jerk are supplied in the
form of 3dimension vectors
giving the angular
acceleration and jerk of the sensor BRF with
respect to the IRF. These vectors are expressed in
thesensorBRF.
NOTE2 InthecaseofexternalinputscomingfromtheS/C
thestarsensorsuppliercanindicatetheminimum
required accuracy for supplied data in order
to
properlyswitchintotracking.
b. A sensor with the autonomous star tracking capability can have the
following additional outputs: measurement of star magnitude for each
trackedstarimage.
C.5 Autonomous attitude determination
a. Asensorwithautonomousattitudedeterminationcanhavethefollowing
additionaloutputs:
1. ameasurementqualityindexorflag,estimatingtheaccuracyofthe
determinedattitude;
2. An inertial angular rate measurement projected on a sensor
definedreferenceframe;
3. a list of the star catalogue numbers for each star
used in the
determination;
4. the position of each star image with respect to a defined sensor
referenceframe;
5. measurementofstarmagnitudeforeachtrackedstarimage.
6. the identification of the optical head(s) used for the attitude
determinationwhenmultipleheadconfigurationisused.
C.6 Autonomous attitude tracking
a. Thefollowingadditionalinputstolaunchtrackingcanbeprovided:
1. theangularaccelerationandjerkofthesensorBRFwithrespectto
theIRF,withtheirvaliditydates;
2. the accuracy of supplied inputs including, in the case of attitude
control,theaccuracyaroundeachaxisofthesensor
BRF.
NOTE1 Angular acceleration and jerk are supplied in the
form of 3dimension vectors giving the angular
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acceleration and jerk of the sensor BRF with
respect to the IRF. These vectors are expressed in
thesensorBRF.
NOTE2 InthecaseofexternalinputscomingfromtheS/C
the star sensor supplier indicates the minimum
required accuracy for supplied data in order to
properlyswitchintotracking.

b. A sensor with autonomous attitude tracking capability can have the
followingadditionaloutputs:
1. ameasurementqualityindexorflagestimatingtheaccuracyofthe
determinedattitude;
2. an angular rate measurement around a sensor defined reference
frame;
3. a list of the star catalogue numbers for each star
used in the
determination;
4. thepositionofeachStarImagewithrespecttoadefinedreference
frame;
5. the identification of the optical head(s) used for the attitude
trackingwhenmultipleheadconfigurationisused.
C.7 Angular rate measurement
a. A sensor with angular rate measurement capability can have the
followingoutputs:
1. ameasurementqualityindexorflag,estimatingtheaccuracyofthe
determinedangularrate;
2. a validity index or flag, estimating the validity of the determined
angularrate.
C.8 Types of star sensors
C.8.1 Star camera
a. A star camera can includethe following additionalcapabilities:(partial)
imagedownload.
C.8.2 Star tracker
a. Astartrackercanincludethefollowingadditionalcapabilities:
1. autonomousstartracking;
2. (partial)imagedownload.
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C.8.3 Autonomous star tracker
a. An autonomous star tracker can include the following additional
capabilities:
1. cartography;
2. startracking;
3. autonomousstar tracking (attitude acquisition with assisted
attitudedetermination);
4. autonomousattitudetracking(withdirectinitialization);
5. angularratemeasurement;
6. (partial)imagedownload.
C.8.4 Summary
a. The specified minimum and additional capabilities for each type of
sensoraresummarizedin
TableC1.
TableC1:Minimumandoptionalcapabilitiesforstar
sensors
Capabilities
Typeofsensor
Cartography
StarTracking
AutonomousStarTracking
AutonomousAttitudeDetermination
AutonomousAttitudeTracking
AngularRateMeasurement
PartialImageDownload
StarCamera
X(X)
StarTracker
X X (X)(X)
AutonomousStarTracker
(X) (X) (X) X X (X) (X)
Key:X=Mandatory,(X)=Optional
TableRows:typeofstarsensors;tablecolumns:capability
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Annex D (informative)
Performance metrics applied to star
sensors
D.1 Overview
Thisannexdiscussestheperformancemetricsusedtoassesstheperformanceof
eachstarsensor capability. The definitions are derivedfromtheESANCR502
(ESA Pointing Error Handbook) taking into account the specific case of star
trackers:
themeasurementerrorsaresmall;
theapproximationofsmallEulerangles
ispossible.
D.2 Application to Star Sensor measurements
D.2.1 Overview
Thisclauseapplies thestandarderrormetricdefinitionstothefollowingtypes
ofStarSensormeasurement:
absoluteratemeasurements;
inertiallyreferencedattitude,viaaquaternion;
singlestarpositionmeasurement.
Thedistinctionbetweenquaternionandstarpositionmeasurementsismade.
D.2.2 Attitude quaternion measurements
The performance metrics AMEq, MMEq, RMEqandMDEq essentiallycapture
the various frequency ranges of measurement
error sources that contribute to
theperformance.Thesearesummarizedin
TableD1.
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TableD1:Measurementerrormetrics
Metric LowerTimePeriodof
ContributionVariation
UpperTimePeriodof
ContributionVariation
AMEq
0
MMEq(tMMEq)
τMMEq
MDEq(tMDEqtOBS,MDEq)
τMDEq τOBS,MDEq
RMEq(tRMEq)
0
τRMEq
Typically,theseperformancemetrics,withappropriatetimeperioddefinitions,
can be used to constrain the following commonly referenced types of
measurementerror:
TotalmeasurementerrorAME
q.
Biaserrors‐MME
q.
Longtermerrorsanddrifts‐MDE
q(withappropriatetimedefinitions).
Shorttermerrors‐MDE
q(withappropriatetimedefinitions).
Noiseerrors,orNoiseEquivalentAngle‐RME
q.
Each of the metrics can be used to constrain rotational or directional errors as
definedinclause
B.5.14.
D.2.3 Star position measurements
The performance metrics AMEs, MMEs, RMEs and MDEs essentially capture
the frequency ranges of measurement
error sources that contribute to the
performance.Thesearesummarizedin
TableD2.
TableD2:StarPositionmeasurementerrormetrics
Metric LowerTimePeriodof
ContributionVariation
UpperTimePeriodof
ContributionVariation
AMES
0
MMES(tMMEs)
τMMEs
MDES(tMDEs,tOBS,MDEs)
τMDEs τOBS,MDEs
RMES(tRMEs)
0
τRMEs
Typically, these metrics, with appropriate time period definitions,can be used
toconstrainthefollowingcommonlyreferencedtypesofmeasurementerror:
TotalmeasurementerrorAMEs.
Biaserrors‐MMEs.
Longtermanddrifterrors‐MDEs(withappropriatetimedefinitions).
Shorttermerrors‐MDEs(withappropriatetime
definitions).
Noiseerrors,orNoiseEquivalentAngle‐RMEs.
Each of the metrics can be used to constrain rotational or directional errors as
definedinclause
B.5.14.
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Annex E (informative)
Statistics
E.1 Confidence level
E.1.1 Overview
The performances have a statistical nature, because they vary with time and
fromonerealizationofasensor to another. Therefore, only an envelope of the
actualperformancescanbespecifiedandprovided.
This envelope is the combination of an upper limit and a performance
confidencelevel.
The performance confidence level
indicates the proportion of the actual
performancesbelowtheupperlimit.
For example, the X absolute measurement error can be 10 arcsec with a
performance confidence level of Pc=95%. This means that the actual errors
fromonesampletoanotherarebelow10arcsecfor95%
ofthecases.
NOTE Performance confidence level is usually 99,7%
(corresponding to a 3 sigma values for Gaussian
distributions).
E.1.2 Accuracy on the confidence level
The verification of the specifications can only be done on a limited set of samples of the
whole statistical population:
Onalimitedtimespan
Onalimitednumberofsensors
The larger the set of samples, the better the knowledge on the performance
confidencelevel(P
c).
This implies that the actual confidence level is not perfectly known, but is
estimated with a certain accuracy ΔP, also called accuracy on the confidence
level.
Thisqualitativenotioncanbemathematicallyexpressedbyusing:
The performance confidence level (P
c): it applies to the performances
quotedbymanufacturersandspecifiedbycustomers(usuallyas3sigma
values).
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And the estimation confidence level. It applies to the estimation of the
performance confidence level (defined above). It represents the
confidencethatthesampleisrepresentativeoftheoverallensemble.
If not specified, confidence level means performance confidence level, and is
denotedP
cinthisdocument.
Theconfidenceestimationaccuracy(
P
Δ
)beingfixed,theminimumnumberof
samples(N)dependsontheestimationconfidencelevel.
For an estimation confidence level 95%, then the minimum number of
samples is given by
2
)1(4
P
PP
N
CC
Δ
= . It means that if the number of
samplesislargerthanN,thentheactualconfidencelevelliesintherange
[]
PPPP
cc
Δ
+
Δ ; in95%ofthecases
Foranestimationconfidence level 99,7%, then the minimum numberof
samplesisgivenby
2
)1(9
P
PP
N
CC
Δ
= . Itmeans ifthenumberofsamples
is larger than N, then the actual confidence level lies in the range
[]
PPPP
cc
Δ
+
Δ ;
in99,7%ofthecases
Furtherdetailscanbefoundinclause
B.2.
NOTE E.g. If the performance confidence level is 99,7%
andthe accuracy is ΔP = 0,1%, thenatleast 11964
samples are considered to actually demonstrate
that the actual performance confidence level is
between99,6%and99,8%(i.e.itisknownwithan
accuracyof
0,1%),withaconfidenceof95%.
E.1.3 Mathematical derivation
Nsamplesofarandomvariablex froma probabilitydistributionfunction p(x)
are considered. Denote the actual performance confidence level of interest
by
C
P , with true value
C
x . Then thenumber of samples
C
N within the set N
lying below
C
x is sampled from a binomial distribution with mean and
variancegivenby:
NPNMean
CC
=)(
NPPNVar
CCC
)1()(
=
Theestimate
C
P
ˆ
oftheperformanceconfidencelevelat
C
x isgivenasfollows:
N
N
P
C
C
=
ˆ
Therefore the mean and variance of the estimate
C
P
ˆ
of the performance
confidencelevelisgivenby:
CC
PPMean =)
ˆ
( (i.e.themeanvalueoftheestimateistheactualvalue)
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N
PP
PVar
CC
C
)1(
)
ˆ
(
=
Now,let
PΔ
betheestimationconfidenceaccuracy,suchthattheactualvalue
P
c of the performance confidence level lies in the range
[
]
PPPP
cc
Δ+Δ
ˆ
;
ˆ
,
withagivenestimationconfidencelevel.
Thevariationsof
C
P
ˆ
aresupposedtofollowaGaussiandistribution.Withthis
assumption,iftheestimationconfidencelevelissetto95%,(whichcorresponds
to
()
C
PVar
ˆ
2± ), then the minimum number of samples in the set N to be
calculatedis:
2
)1(4
P
PP
N
CC
Δ
=
For a 99,7% estimation confidence level on
C
P
ˆ
, the formula becomes
2
)1(9
P
PP
N
CC
Δ
=
, because 99,7% corresponds to a 3 sigma value for a
Gaussiandistribution.
Moregenerally,
2
2
)1(
P
PPn
N
CCC
Δ
=
foranCsigma estimation confidence level
of a Gaussian distribution.
NOTE For example, if the performance confidence level
on the error is 99,7% and the accuracy is
ΔP=0,1%, then at least 11964 samples are the
minimum number of samples used to actually
demonstrate that the actual confidence level is
between99,6%and
99,8%(i.e.itisknownwithan
accuracy of 0,1%), with an estimation confidence
levelof95%.
E.1.4 Minimum number of runs with no failure
ThepreviousclausefocusesontheminimumnumberNofsimulationstorunto
demonstratetheperformanceswithinagivenperformanceconfidenceleveland
agivenaccuracyontheestimationconfidencelevel.
Anotherapproach, more efficient from the implementation point ofview,isto
considerthenumberN
t
ofsimulationstorunifnofailureoccurstodemonstrate
the same performances. In this context, a failure is a simulation in which the
performanceleveltobedemonstratedisexceeded.
ThisnumberofsimulationsN
t
isusuallymuchsmallerthanN,whichmakesthe
approachmoreappropriate.
NOTE E.g.iftherequirementisspecifiedat99,73%,then
the number of samplesto estimate this
performance confidence level with a 95%
estimation confidence of the real value being
within ±0,1% of the
estimate is N=11964.
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However assuming no failures are seen, only
N
t
=1108 runs are required to prove that the
probability of failure is <0,27% to a 95%
estimationconfidencelevel.
Suppose that we have a specification P(x<x
max)>Pmin, with some statistical
interpretation. This means that for the system to meet its specification, in any
given trial the probability of having x<x
max is at least Pmin. Equivalently, the
probabilityofhavingafailure(x>x
max)inanygiventrialislessthanPf
max
=1Pmin.
Given a Monte Carlo campaign with N
t
runs, ofwhich nf of these are failures.
Assuming that the real underlying probability of failure (not known to the
experimenter) is P
f, then the probability of observing nf failures in N
t
trials is
givenbythebinomialformula:
(
)
()
(
)
f
f
nN
f
n
f
ff
ff
PP
nNn
N
NPnP
-
-1
!-!
!
,|
=
The relation
()
(
)
()
()
ABP
BP
AP
BAP || = with the normalization condition
()
= 1| dABAP
yields:
()()
(
)
(
)
1-1
1
!-!
!
1
0
-
=
dPPPPP
nPnNn
N
f
f
nN
f
n
ff
fff
Ifthereisnoaprioriinformationabouttheprobabilityoffailure,thenthemost
conservationapproach is toassumethatthe probability offailureis uniformly
distributedbetween0and1:
(
)
1
=
f
PP
Thisyields
()()
(
)
1-1
1
!-!
!
1
0
-
=
dPPP
nPnNn
N
f
f
nN
f
n
f
fff
Then,foragivennumberofobserved failuresn
ftheprobabilitydistributionof
P
fisfoundtobe:
()
(
)
()
1
0
-
-
-1
-1
,|
dPPP
PP
NnPP
f
f
f
f
nN
n
nN
f
n
f
ff
=
IftheprobabilityoffailureislessthansomevalueP
f
max
,thespecificationismet.
(This is equivalent to a minimum probability of not failing the specification.)
Givenn
ffailuresin
N
trials,the confidenceofthe specificationactuallybeing
met(i.e.ofP
freallybeinglessthanPf
max
)is:
()
(
)
()
()
1-,1,
-1
-1
max
1
0
-
0
-
max
max
+
+=
==
fffinc
nN
n
P
nN
n
ff
nNnP
dPPP
dPPP
PPprobC
f
f
f
f
f
β
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whereβincistheincompletebetafunctiongivenby:
()
()
()
=
1
0
1
1
0
1
1
1
1
,,
dttt
dttt
bax
b
a
x
b
a
inc
β
Thisfunctionisavailableinusualengineeringtools.
Usingthis formula,itis possibleto work outthe minimum numberof runs in
ordertomeetthespecificationswithagivenprobabilitytoagivenperformance
confidencelevel.
TableE1givesnumericalapplicationsforvariouscases.
TableE1:Minimumnumberofsimulationstoverifya
performanceatperformanceconfidencelevelP
Ctoan
estimationconfidencelevelof95%
MinimumnumberofrunsfornumberofobservedfailuresN
fail
Performance
confidencelevelP
C
N
fail=0 Nfail=1 Nfail=2 Nfail=3
68% 7 12 17 21
95% 58 92 123 152
99,73% 1108 1755 2329 2869
NOTE:‘failure’inthiscontextmeansviolationofthespecifiedbound,x>xmax.
There is no equivalenceto the estimation confidence accuracy
PΔ introduced
inclauseB.1.3.Itmeansthattheestimationconfidencelevelisatleastthelevel
specified(e.g.95%inthetableabove).
E.2 Statistical interpretation of measurement error
metrics
EachofthemetricsdefinedinclauseB.5istypicallyspecifiedandusedwithan
associatedconfidencelevel.
Anyperformancemetricsdependsonseveralvariables:
thetimet;
therealizationofthesensor(involvingthemanufacturingprocess);
the observation conditions in which the performances are obtained
(e.g.angular rate applied on the sensors, orientation with
respect to the
celestialvault).
Asitisnotpossibletobuildarepresentativesamplesetofsensors,thenotionof
statistical
ensemble is used. A statistical ensemble of sensors is defined as a
collectionofsensorsrepresentative ofthemanufacturing process, in whichnot
allsensorsarenecessarilybuilt.
Because a metrics depends on several variables, there are several ways to
interpretaspecificationanditsconfidencelevel:
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Temporalinterpretation
The worst case combination of sensors and observations is
considered.
The worstcase sensor/observation combination is defined as the
worstcase sensor observing the worstcase direction in the
celestial vault under the worstcase observation conditions. The
worstcasedirectionistheoneleadingtotheworstperformanceof
the sensor. It is related to the worst distribution of stars
over the
starsensorfieldofview,takingintoaccountembeddedalgorithms
andcatalogues.
Theperformancesareestablishedwithrespecttotime.
The specification metric is ‘less than S for n% of the time for a
worstcase sensor/observation from a
statistical ensemble of
sensors/observations’.
Ensembleinterpretation
Astatisticalcollectionofsensorsisarbitrarilychosen.
Agivensetofobservationsisarbitrarilychosen.
The time is set to the worst case time, i.e. when the performances
obtainedforagivensensorandobservationareworst.
Thespecificationmetricforthistype ofvariabilityis‘lessthanthe
level S in confidence level n% of a
statistical ensemble of
sensors/observationsfortheworstcasetime’.
Mixedinterpretation
The mixed interpretation combines the ensemble and temporal
variationtocapturetheerrorvariabilitybothovertimeandacross
theensemble.
Thespecificationmetricforthistypeofvariabilityis‘forarandom
sensor/observation from the statistical
ensemble,and at a random
time,themetricislessthanSwithaprobabilityofn%’.
For a generic measurement error source with an amplitude and a time
variation, the ensemble interpretation gives the distribution of the error
amplitude over the statistical
ensemble of sensors/observations, while the
temporal interpretation covers the error variation over time for the worstcase
amplitude.
For the AME, RME and MDE metrics defined in clause
B.5, the statistical
interpretation can in principle be ensemble, temporal or mixed. However, the
nature of the MME metric means that only an ensemble interpretation is
appropriate. Specific identification of the interpretations to be used in this
specificationisgivenin
AnnexD.
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Annex F (informative)
Transformations between coordinate
frames
Transformationsbetweenanytwocoordinateframes,AandBcanbedescribed
by the transformation matrix
BA
T
which transforms the components of a
vectorfrom‘B’frameto‘Aframe:
B
BA
A
rTr
=
where
A
r arethecomponentsofthevector r inthe‘A’frame,and
B
r arethe
componentsofthesamevector
r inthe‘B’frame.
The discrepancybetween both frames ‘A’ and ‘B’ is defined by 3 Euler angles
around 3 distinct axes. In this Standard, the rotations are always small,
thereforetheorder of the rotations is notimportantandthese rotations canbe
takentoberotationsaroundthe
X,Y‐andZaxesofeitherframe.
Thetransformationissimply:
φΔθΔ
φΔψΔ
θΔψΔ
1
1
1
BA
T
where
φ
Δ ,
θ
Δ
and
ψ
Δ
are the 3 small rotations respectively around X, Y
andZaxestransformingthe‘B’frameintothe‘A’frame.
Thediscrepancybetweenbothframes‘A’and‘B’is:
Δ
Δ
Δ
=
ψ
θ
φ
ε
Thediscrepancyisafunctionofthetime.
NOTE Theperformancesofstarsensorsaremeasuredby
applying the metrics defined in
Annex D to this
vector
ε
.
For star sensors, this vector typically represents the angular errors between a
measuredquantityanditsactualvalue.
NOTE E.g. With ‘A’ frame being the actual star sensor
frame and ‘B’ frame being the measured star
ECSSEST6020CRev.1
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81
sensorframe,then
ε
represents themeasurement
errorsofthestarsensor(see
FigureF1).
B
y
B
y
B
z
B
y
B
x
1
st
rotation
Original Frame
B
z
B
x
After 1
st
rotation
2
nd
rotation
After 2
nd
rotation
B
z
B
y
B
x
B
z
B
x
Final Frame
A
z
A
x
A
y
FigureF1:Anglerotationsequence
In this case the 3axisEulerrotation parameterization corresponds to rotations
aroundtheBframeaxes.
The separation of two frames A and B, defined in the ESA Pointing Error
Handbookandwrittenas
(
)
BA
Tsep
isdefinedas:
()
Δ
Δ
Δ
==
ψ
θ
φ
ε
BA
Tsep
Thisfunctionrepresentsthediscrepancybetweenthetwoframesandisusedto
measurethestarsensorperformances.
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Annex G (informative)
Contributing Error Sources
G.1 Overview
ThisannexlinkstheerrorcontributorstothedefinitionsderivedfromtheESA
NCR502 (ESA Pointing Error Handbook). The traditional contributors and
performances are compared with generalized error with respect to the
correspondingcorrelationtime
τgivenforeachcontributor.
TableG1:Contributingerrorsources
Errorcontributors Comments
Bias
‐ ongroundcalibrationresidual
‐ launchinducedmisalignment(vibrations,
depressurization,gravity…)
BRFvsMRFmisalignmentduetoafterlaunch
ageing
MME(
τ=infinite)
MME(
τ=lifetime)
Thermoelasticerror
BRFvsMRFstabilitydueto:
‐ stabilizedopticalheadtemperature
‐ gradientcausedbyconductiveand
radiativeeffects
MDE(
τ=oncethethermalscenarioisknown.)
τ =correlationlength
τ obs=observationlength
FOVspatialerrors
‐ PointSpreadFunctionvariabilityacross
theFOV
‐ residualofcalibrationoffocallength
(includingitstemperaturesensibility)andoptical
distortions(includingchromatism)
‐ residualofaberrationoflightincase
whereitiscorrectedatquaternionlevelandnotat
starlevel
‐ CCDCTEeffect(includingits
degradationsdueto
radiations)
‐ catalogueerror(includingstarproper
motionandparallax)
Theamplitudeoftheseerrorsareindependentofthe
rate.
The
τisassessedbythesupplierintheangular
domain.
Thereisaneedtogetthefiguresforseveral
τ values.Theuseofautocorrelationfunctionof
spatialerrorisrecommended.
MDE(
τtobedescribed)
Canbeconvertedbytheuserintimedomain
dependingonthespecificapplication.usingangular
rate
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Pixelspatialerrors
‐ detectornonuniformity(FPN,DSNU
(DS(T),radiation,integrationtime…),
PRNU(straylight,starsignalphotonicnoise)…)
‐ centroiding(ratedependent)
The
τisassessedbythesupplierintheangular
domain.
Canbeconvertedbytheuserintimedomain
dependingonthespecificapplicationusingangular
rate.
MDE(
τlinkedtopixelFOV)
Temporalnoise
‐ starsignalshotnoisedependingonstar
signal(StarMagnitude,exposuretime,optical
contamination,transmissionloss,defocus,rate…)
‐ backgroundsignalshotnoise(straylight
level,detectortemperature…)
‐ readoutnoise
‐ quantificationnoise
‐ datationnoise
RME(
τ=0orlessthanthesampletime)
Aberrationoflightorresidualofaberrationof
lightcorrectionifcorrectedatstarlevel
MDE(
τ=TBDbyuser)
residualofaberrationoflightcorrectionifcorrected
atstarlevel
Asthiserrorisverydeterministic,itispossibleto
correctitinsidethestartracker‐supposingthatthe
velocityinformationisgiventothestartracker.A
fewcasesarequoted:
1) acorrectionis
performedforeverystar
direction,
2) auniquecorrectionisperformedgloballyfor
auniquedirection(example:lineofsight,or
barycentreofthemeasuredstars)andappliedonthe
quaternionoroneachstarmeasurement,
3) acorrectionisperformedonlyfortheEarth/
Sunvelocity,
4) nocorrection
isperformed.
Dependingonthecorrection,theerrorresidualis:
‐ aFOVspatialerrorifthecorrectionis
performedglobally(case2)
‐ anorbitalerrorinthecase3(dependingalso
ontheattitudeofthespacecraft)
‐ alongtermerror(oneyear)+orbitalerror
forthecase
4.
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Annex H (informative)
Example of data sheet
H.1 Introduction
The data sheet in Figure H1 shows an example of data sheetfor autonomous
startracker.
Thefieldsthatcanbefilledinareidentifiedinanitalicfont.
Theexamplevalues filledin arejustforformattingpurposesanddonotrelate
toanexistingstarsensor.
H.2 Rules applied
ThefollowingruleshavebeenappliedtoprovidethedatasheetinFigureH1:
useofthecontentoftheexampledatasheetproposedinthe“StarSensor
Terminology and Performance Specification Standard”, issue 1 and
additionofsomekeyitems.(firstversionofthepresentdocumentissued
byESAstudies);
the data sheet has been limited to one page
of format A4 but is not
mandatory.
ECSSEST6020CRev.1
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85
Detailed Data Sheet
Companies Logo
Name: Name as supplied by manufacturer
Type:
Autonomous Star Tracker
photograph of the unit
Configuration:
Single Box, Single Head
Specification: Detector:
STAR1000 APS
FOV:
20 x 30 deg rectangular
Interface:
MIL-1553
Power:
<10,0 W
Voltage
V
Output Frequency:
10 Hz
T
operational
:
-10/ +20
°C
Mass (Baffle):
1,0 (1,0) kg
Dimension:
120 120 225
mm
(including baffle) width depth height
EOL performances mission dependent
BOL performance
quoted @ worst case temperature / celestial bodies aspect angles
quoted for normal to LOS and for along LOS
Aspect Angles:
Sun Exclusion Angle:
25 deg (@ 1AU)
Earth Exclusion Angle:
20 deg (illuminated Earth limb at 70 km)
Moon Exclusion Angle:
n/a deg (full Moon accepted in the FOV)
(normal to / along LOS) statistical distribution
Bias
5,0 / 15 arcsec max
TBD
thermal stability
0,1 arcsec/K
uniform
spatial error (Pixel)
2,0 / 4,0
arcsec p/p arcsec TBD
spatial error (FOV)
1,0 / 3,0 arcsec p/p
TBD
Attitude tracking can be performed under the following conditions:
statistical distribution
Rate
0,0 1,0 5,0
deg/sec
Acceleration
0,0 0,1 1,0
deg/sec
2
temporal noise
10 / 70 14 / 98 25 / 175
arcsec 3
σ
gaussian
Prob. of correct
0,9999
0,9990
0,9500
P (10s, random directions)
Prob. of false attitude
0,0010
0,0030
0,0100
P (10s, random directions)
Maintenance level of 999s/1000s 990s/1000s 999s/1000s
# of false stars
0 10 50
P (10s, random directions)
0,9999
0,9950
0,7000
Prob. of correct attitude determination
P (10s, random directions)
0,0010
0,0050
0,0100
Prob. of false attitude determination
Track
Maintenance
/Period
999s/1000s 990s/1000s 999s/1000s Maintenance level of tracking
S
p
ecial Features:
Tolerance u
p
to 1000 SET/cm
2
/s at initial ac
q
uisition
Tolerance up to 4000 SET/cm
2
/s in tracking
Life time 15 years geo-synchronous orbit (FITS number)
High-rel EEE-parts, ITAR free,
100krad
Attitude quaternion
Angular rate and Acceleration limit for tracking : 1 °/s and 3° / s
2
Time stamp
Health status and star
p
osition & ma
g
nitude
Image download / Code&Data down- & up- load
FigureH1:Exampleofdetaileddatasheet
ECSSEST6020CRev.1
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86
Bibliography
ECSSSST00 ECSSsystemDescription,implementationand
generalrequirements
ESA-NCR-502
ESAPointingErrorHandbook